poimirlem17 - Mathbox for Brendan Leahy

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Theorem poimirlem17 35074

Description: Lemma for poimir 35090 establishing existence for poimirlem18 35075. (Contributed by Brendan Leahy, 21-Aug-2020.)

**Hypotheses**
Ref	Expression
poimir.0	⊢ (𝜑 → 𝑁 ∈ ℕ)
poimirlem22.s	⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem22.1	⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑_m (1...𝑁)))
poimirlem22.2	⊢ (𝜑 → 𝑇 ∈ 𝑆)
poimirlem18.3	⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾)
poimirlem18.4	⊢ (𝜑 → (2^nd ‘𝑇) = 0)

**Assertion**
Ref	Expression
poimirlem17	⊢ (𝜑 → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)

Distinct variable groups: 𝑓,𝑗,𝑛,𝑝,𝑡,𝑦,𝑧 𝜑,𝑗,𝑛,𝑦 𝑗,𝐹,𝑛,𝑦 𝑗,𝑁,𝑛,𝑦 𝑇,𝑗,𝑛,𝑦 𝜑,𝑝,𝑡 𝑓,𝐾,𝑗,𝑛,𝑝,𝑡 𝑓,𝑁,𝑝,𝑡 𝑇,𝑓,𝑝 𝜑,𝑧 𝑓,𝐹,𝑝,𝑡,𝑧 𝑧,𝐾 𝑧,𝑁 𝑡,𝑇,𝑧 𝑆,𝑗,𝑛,𝑝,𝑡,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑓) 𝑆(𝑓) 𝐾(𝑦)

**Proof of Theorem poimirlem17**
Step	Hyp	Ref	Expression
1		poimir.0	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		poimirlem22.s	. . . . 5 ⊢ 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
3		poimirlem22.1	. . . . 5 ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑_m (1...𝑁)))
4		poimirlem22.2	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
5		poimirlem18.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾)
6		poimirlem18.4	. . . . 5 ⊢ (𝜑 → (2^nd ‘𝑇) = 0)
7	1, 2, 3, 4, 5, 6	poimirlem16 35073	. . . 4 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))
8		elfznn0 12995	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ₀)
9	8	nn0red 11944	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
10	9	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ)
11	1	nnzd 12074	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ ℤ)
12		peano2zm 12013	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
13	11, 12	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
14	13	zred 12075	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
15	14	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ)
16	1	nnred 11640	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℝ)
17	16	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℝ)
18		elfzle2 12906	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑁 − 1))
19	18	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ≤ (𝑁 − 1))
20	16	ltm1d 11561	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
21	20	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) < 𝑁)
22	10, 15, 17, 19, 21	lelttrd 10787	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
23	22	adantlr 714	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
24		fveq2 6645	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ → (2^nd ‘𝑡) = (2^nd ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩))
25		opex 5321	. . . . . . . . . . . . . . . 16 ⊢ ⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩ ∈ V
26		op2ndg 7684	. . . . . . . . . . . . . . . 16 ⊢ ((⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩ ∈ V ∧ 𝑁 ∈ ℕ) → (2^nd ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) = 𝑁)
27	25, 1, 26	sylancr 590	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) = 𝑁)
28	24, 27	sylan9eqr 2855	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (2^nd ‘𝑡) = 𝑁)
29	28	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘𝑡) = 𝑁)
30	23, 29	breqtrrd 5058	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < (2^nd ‘𝑡))
31	30	iftrued 4433	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = 𝑦)
32	31	csbeq1d 3832	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋𝑦 / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
33		vex 3444	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
34		oveq2 7143	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦))
35	34	imaeq2d 5896	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑦 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑦)))
36	35	xpeq1d 5548	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑦 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑦)) × {1}))
37		oveq1 7142	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1))
38	37	oveq1d 7150	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁))
39	38	imaeq2d 5896	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑦 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)))
40	39	xpeq1d 5548	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑦 → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)) × {0}))
41	36, 40	uneq12d 4091	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑦 → ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)) × {0})))
42	41	oveq2d 7151	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑦 → ((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)) × {0}))))
43	33, 42	csbie 3863	. . . . . . . . . . . 12 ⊢ ⦋𝑦 / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)) × {0})))
44		2fveq3 6650	. . . . . . . . . . . . . 14 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩)))
45		op1stg 7683	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩ ∈ V ∧ 𝑁 ∈ ℕ) → (1^st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩)
46	25, 1, 45	sylancr 590	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1^st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩)
47	46	fveq2d 6649	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1^st ‘(1^st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩)) = (1^st ‘⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩))
48		ovex 7168	. . . . . . . . . . . . . . . . 17 ⊢ (1...𝑁) ∈ V
49	48	mptex 6963	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∈ V
50		fvex 6658	. . . . . . . . . . . . . . . . 17 ⊢ (2^nd ‘(1^st ‘𝑇)) ∈ V
51	48	mptex 6963	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∈ V
52	50, 51	coex 7617	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) ∈ V
53	49, 52	op1st 7679	. . . . . . . . . . . . . . 15 ⊢ (1^st ‘⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩) = (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0)))
54	47, 53	eqtrdi 2849	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1^st ‘(1^st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩)) = (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))))
55	44, 54	sylan9eqr 2855	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (1^st ‘(1^st ‘𝑡)) = (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))))
56		fveq2 6645	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ → (1^st ‘𝑡) = (1^st ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩))
57	56, 46	sylan9eqr 2855	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (1^st ‘𝑡) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩)
58	57	fveq2d 6649	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩))
59	49, 52	op2nd 7680	. . . . . . . . . . . . . . . . 17 ⊢ (2^nd ‘⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩) = ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
60	58, 59	eqtrdi 2849	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (2^nd ‘(1^st ‘𝑡)) = ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))))
61	60	imaeq1d 5895	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑦)) = (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)))
62	61	xpeq1d 5548	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑦)) × {1}) = ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}))
63	60	imaeq1d 5895	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)) = (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))
64	63	xpeq1d 5548	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)) × {0}) = ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))
65	62, 64	uneq12d 4091	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)) × {0})) = (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))
66	55, 65	oveq12d 7153	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → ((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑦)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑦 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))
67	43, 66	syl5eq 2845	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → ⦋𝑦 / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))
68	67	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋𝑦 / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))
69	32, 68	eqtrd 2833	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))
70	69	mpteq2dva 5125	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))
71	70	eqeq2d 2809	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))))
72	71	ex 416	. . . . . 6 ⊢ (𝜑 → (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))))
73	72	alrimiv 1928	. . . . 5 ⊢ (𝜑 → ∀𝑡(𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))))
74		oveq2 7143	. . . . . . . . . . 11 ⊢ (1 = if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 1) = (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0)))
75	74	eleq1d 2874	. . . . . . . . . 10 ⊢ (1 = if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0) → ((((1^st ‘(1^st ‘𝑇))‘𝑛) + 1) ∈ (0..^𝐾) ↔ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0)) ∈ (0..^𝐾)))
76		oveq2 7143	. . . . . . . . . . 11 ⊢ (0 = if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 0) = (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0)))
77	76	eleq1d 2874	. . . . . . . . . 10 ⊢ (0 = if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0) → ((((1^st ‘(1^st ‘𝑇))‘𝑛) + 0) ∈ (0..^𝐾) ↔ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0)) ∈ (0..^𝐾)))
78		fveq2 6645	. . . . . . . . . . . . . 14 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → ((1^st ‘(1^st ‘𝑇))‘𝑛) = ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)))
79	78	oveq1d 7150	. . . . . . . . . . . . 13 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 1) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))
80	79	adantl 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 1) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))
81		elrabi 3623	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
82	81, 2	eleq2s 2908	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
83		xp1st 7703	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
84	4, 82, 83	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
85		xp1st 7703	. . . . . . . . . . . . . . . . 17 ⊢ ((1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
86		elmapi 8411	. . . . . . . . . . . . . . . . 17 ⊢ ((1^st ‘(1^st ‘𝑇)) ∈ ((0..^𝐾) ↑_m (1...𝑁)) → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
87	84, 85, 86	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
88	4, 82	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
89		xp2nd 7704	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1^st ‘𝑇) ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
90	88, 83, 89	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
91		f1oeq1 6579	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (2^nd ‘(1^st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
92	50, 91	elab 3615	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
93	90, 92	sylib 221	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
94		f1of 6590	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)⟶(1...𝑁))
95	93, 94	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)⟶(1...𝑁))
96		nnuz 12269	. . . . . . . . . . . . . . . . . . 19 ⊢ ℕ = (ℤ_≥‘1)
97	1, 96	eleqtrdi 2900	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
98		eluzfz1 12909	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 1 ∈ (1...𝑁))
99	97, 98	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 1 ∈ (1...𝑁))
100	95, 99	ffvelrnd 6829	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (1...𝑁))
101	87, 100	ffvelrnd 6829	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ (0..^𝐾))
102		elfzonn0 13077	. . . . . . . . . . . . . . 15 ⊢ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ (0..^𝐾) → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ ℕ₀)
103		peano2nn0 11925	. . . . . . . . . . . . . . 15 ⊢ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ ℕ₀ → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ∈ ℕ₀)
104	101, 102, 103	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ∈ ℕ₀)
105		elfzo0 13073	. . . . . . . . . . . . . . . 16 ⊢ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ (0..^𝐾) ↔ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ ℕ₀ ∧ 𝐾 ∈ ℕ ∧ ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) < 𝐾))
106	101, 105	sylib 221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ ℕ₀ ∧ 𝐾 ∈ ℕ ∧ ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) < 𝐾))
107	106	simp2d 1140	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐾 ∈ ℕ)
108		elfzolt2 13042	. . . . . . . . . . . . . . . . 17 ⊢ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ (0..^𝐾) → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) < 𝐾)
109	101, 108	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) < 𝐾)
110	101, 102	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ ℕ₀)
111	110	nn0zd 12073	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ ℤ)
112	107	nnzd 12074	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐾 ∈ ℤ)
113		zltp1le 12020	. . . . . . . . . . . . . . . . 17 ⊢ ((((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) ∈ ℤ ∧ 𝐾 ∈ ℤ) → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) < 𝐾 ↔ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ≤ 𝐾))
114	111, 112, 113	syl2anc 587	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) < 𝐾 ↔ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ≤ 𝐾))
115	109, 114	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ≤ 𝐾)
116		fvex 6658	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ V
117		eleq1 2877	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → (𝑛 ∈ (1...𝑁) ↔ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (1...𝑁)))
118	117	anbi2d 631	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ↔ (𝜑 ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (1...𝑁))))
119		fveq2 6645	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → (𝑝‘𝑛) = (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)))
120	119	neeq1d 3046	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → ((𝑝‘𝑛) ≠ 𝐾 ↔ (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) ≠ 𝐾))
121	120	rexbidv 3256	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → (∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾 ↔ ∃𝑝 ∈ ran 𝐹(𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) ≠ 𝐾))
122	118, 121	imbi12d 348	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1) → (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 𝐾) ↔ ((𝜑 ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) ≠ 𝐾)))
123	116, 122, 5	vtocl 3507	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) ≠ 𝐾)
124	100, 123	mpdan 686	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∃𝑝 ∈ ran 𝐹(𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) ≠ 𝐾)
125		fveq1 6644	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘1)))
126	87	ffnd 6488	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
127	126	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1^st ‘(1^st ‘𝑇)) Fn (1...𝑁))
128		1ex 10626	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 1 ∈ V
129		fnconstg 6541	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (1 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))
130	128, 129	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1)))
131		c0ex 10624	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 0 ∈ V
132		fnconstg 6541	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (0 ∈ V → (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
133	131, 132	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))
134	130, 133	pm3.2i 474	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
135		dff1o3 6596	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ^◡(2^nd ‘(1^st ‘𝑇))))
136	135	simprbi 500	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡(2^nd ‘(1^st ‘𝑇)))
137	93, 136	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → Fun ^◡(2^nd ‘(1^st ‘𝑇)))
138		imain 6409	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (Fun ^◡(2^nd ‘(1^st ‘𝑇)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
139	137, 138	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
140		nn0p1nn 11924	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ ℕ₀ → (𝑦 + 1) ∈ ℕ)
141	8, 140	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ)
142	141	nnred 11640	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ)
143	142	ltp1d 11559	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) < ((𝑦 + 1) + 1))
144		fzdisj 12929	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
145	144	imaeq2d 5896	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2^nd ‘(1^st ‘𝑇)) “ ∅))
146		ima0 5912	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ∅) = ∅
147	145, 146	eqtrdi 2849	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
148	143, 147	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
149	139, 148	sylan9req 2854	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅)
150		fnun 6434	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
151	134, 149, 150	sylancr 590	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
152		imaundi 5975	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
153	141	peano2nnd 11642	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℕ)
154	153, 96	eleqtrdi 2900	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ_≥‘1))
155	154	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) + 1) ∈ (ℤ_≥‘1))
156	1	nncnd 11641	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝜑 → 𝑁 ∈ ℂ)
157		npcan1 11054	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
158	156, 157	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
159	158	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
160		elfzuz3 12899	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ_≥‘𝑦))
161		eluzp1p1 12258	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
162	160, 161	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
163	162	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑦 + 1)))
164	159, 163	eqeltrrd 2891	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ_≥‘(𝑦 + 1)))
165		fzsplit2 12927	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑦 + 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘(𝑦 + 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
166	155, 164, 165	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
167	166	imaeq2d 5896	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))))
168		f1ofo 6597	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
169		foima 6570	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
170	93, 168, 169	3syl 18	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
171	170	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
172	167, 171	eqtr3d 2835	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
173	152, 172	syl5eqr 2847	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
174	173	fneq2d 6417	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∪ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ↔ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁)))
175	151, 174	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
176	48	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V)
177		inidm 4145	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
178		eqidd 2799	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) = ((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)))
179		f1ofn 6591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
180	93, 179	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
181	180	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
182		fzss2 12942	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
183	164, 182	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
184	141, 96	eleqtrdi 2900	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ_≥‘1))
185		eluzfz1 12909	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑦 + 1) ∈ (ℤ_≥‘1) → 1 ∈ (1...(𝑦 + 1)))
186	184, 185	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ (1...(𝑦 + 1)))
187	186	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ (1...(𝑦 + 1)))
188		fnfvima 6973	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁) ∧ (1...(𝑦 + 1)) ⊆ (1...𝑁) ∧ 1 ∈ (1...(𝑦 + 1))) → ((2^nd ‘(1^st ‘𝑇))‘1) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))
189	181, 183, 187, 188	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘(1^st ‘𝑇))‘1) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))
190		fvun1 6729	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∧ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) ∧ ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘1)) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2^nd ‘(1^st ‘𝑇))‘1)))
191	130, 133, 190	mp3an12 1448	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) ∩ ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1)))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘1)) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2^nd ‘(1^st ‘𝑇))‘1)))
192	149, 189, 191	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘1)) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2^nd ‘(1^st ‘𝑇))‘1)))
193	128	fvconst2 6943	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((2^nd ‘(1^st ‘𝑇))‘1) ∈ ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2^nd ‘(1^st ‘𝑇))‘1)) = 1)
194	189, 193	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2^nd ‘(1^st ‘𝑇))‘1)) = 1)
195	192, 194	eqtrd 2833	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘1)) = 1)
196	195	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (1...𝑁)) → (((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2^nd ‘(1^st ‘𝑇))‘1)) = 1)
197	127, 175, 176, 176, 177, 178, 196	ofval 7398	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2^nd ‘(1^st ‘𝑇))‘1) ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))
198	100, 197	mpidan 688	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))
199	125, 198	sylan9eqr 2855	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))
200	199	adantllr 718	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑝 ∈ ran 𝐹) ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))
201		fveq2 6645	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑡 = 𝑇 → (2^nd ‘𝑡) = (2^nd ‘𝑇))
202	201	breq2d 5042	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑡 = 𝑇 → (𝑦 < (2^nd ‘𝑡) ↔ 𝑦 < (2^nd ‘𝑇)))
203	202	ifbid 4447	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)))
204	203	csbeq1d 3832	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
205		2fveq3 6650	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑡 = 𝑇 → (1^st ‘(1^st ‘𝑡)) = (1^st ‘(1^st ‘𝑇)))
206		2fveq3 6650	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑡 = 𝑇 → (2^nd ‘(1^st ‘𝑡)) = (2^nd ‘(1^st ‘𝑇)))
207	206	imaeq1d 5895	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)))
208	207	xpeq1d 5548	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}))
209	206	imaeq1d 5895	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑡 = 𝑇 → ((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
210	209	xpeq1d 5548	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ (𝑡 = 𝑇 → (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
211	208, 210	uneq12d 4091	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑡 = 𝑇 → ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
212	205, 211	oveq12d 7153	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑡 = 𝑇 → ((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
213	212	csbeq2dv 3835	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
214	204, 213	eqtrd 2833	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
215	214	mpteq2dv 5126	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
216	215	eqeq2d 2809	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
217	216, 2	elrab2 3631	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑇 ∈ 𝑆 ↔ (𝑇 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
218	217	simprbi 500	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑇 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
219	4, 218	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
220	219	rneqd 5772	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ran 𝐹 = ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
221	220	eleq2d 2875	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑝 ∈ ran 𝐹 ↔ 𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
222		eqid 2798	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
223		ovex 7168	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
224	223	csbex 5179	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) ∈ V
225	222, 224	elrnmpti 5796	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
226	221, 225	syl6bb 290	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑝 ∈ ran 𝐹 ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
227	6	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘𝑇) = 0)
228		elfzle1 12905	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑦 ∈ (0...(𝑁 − 1)) → 0 ≤ 𝑦)
229	228	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → 0 ≤ 𝑦)
230	227, 229	eqbrtrd 5052	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (2^nd ‘𝑇) ≤ 𝑦)
231		0re 10632	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ 0 ∈ ℝ
232	6, 231	eqeltrdi 2898	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℝ)
233		lenlt 10708	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((2^nd ‘𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2^nd ‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2^nd ‘𝑇)))
234	232, 9, 233	syl2an 598	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ((2^nd ‘𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2^nd ‘𝑇)))
235	230, 234	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑦 < (2^nd ‘𝑇))
236	235	iffalsed 4436	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1))
237	236	csbeq1d 3832	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋(𝑦 + 1) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
238		ovex 7168	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 + 1) ∈ V
239		oveq2 7143	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1)))
240	239	imaeq2d 5896	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 = (𝑦 + 1) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))))
241	240	xpeq1d 5548	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 = (𝑦 + 1) → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) = (((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}))
242		oveq1 7142	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1))
243	242	oveq1d 7150	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁))
244	243	imaeq2d 5896	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 = (𝑦 + 1) → ((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
245	244	xpeq1d 5548	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑗 = (𝑦 + 1) → (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
246	241, 245	uneq12d 4091	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑗 = (𝑦 + 1) → ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
247	246	oveq2d 7151	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑗 = (𝑦 + 1) → ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
248	238, 247	csbie 3863	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ⦋(𝑦 + 1) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
249	237, 248	eqtrdi 2849	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
250	249	eqeq2d 2809	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑦 ∈ (0...(𝑁 − 1))) → (𝑝 = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) ↔ 𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))))
251	250	rexbidva 3255	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ⦋if(𝑦 < (2^nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))))
252	226, 251	bitrd 282	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑝 ∈ ran 𝐹 ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))))
253	252	biimpa 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑝 ∈ ran 𝐹) → ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ((1^st ‘(1^st ‘𝑇)) ∘_f + ((((2^nd ‘(1^st ‘𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2^nd ‘(1^st ‘𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
254	200, 253	r19.29a 3248	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑝 ∈ ran 𝐹) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))
255		eqtr3 2820	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ∧ 𝐾 = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1)) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = 𝐾)
256	255	ex 416	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) → (𝐾 = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = 𝐾))
257	254, 256	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑝 ∈ ran 𝐹) → (𝐾 = (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) → (𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) = 𝐾))
258	257	necon3d 3008	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑝 ∈ ran 𝐹) → ((𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) ≠ 𝐾 → 𝐾 ≠ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1)))
259	258	rexlimdva 3243	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (∃𝑝 ∈ ran 𝐹(𝑝‘((2^nd ‘(1^st ‘𝑇))‘1)) ≠ 𝐾 → 𝐾 ≠ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1)))
260	124, 259	mpd 15	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ≠ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))
261	104	nn0red 11944	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ∈ ℝ)
262	107	nnred 11640	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ ℝ)
263	261, 262	ltlend 10774	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) < 𝐾 ↔ ((((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ≤ 𝐾 ∧ 𝐾 ≠ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1))))
264	115, 260, 263	mpbir2and 712	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) < 𝐾)
265		elfzo0 13073	. . . . . . . . . . . . . 14 ⊢ ((((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ∈ (0..^𝐾) ↔ ((((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ∈ ℕ₀ ∧ 𝐾 ∈ ℕ ∧ (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) < 𝐾))
266	104, 107, 264, 265	syl3anbrc 1340	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ∈ (0..^𝐾))
267	266	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → (((1^st ‘(1^st ‘𝑇))‘((2^nd ‘(1^st ‘𝑇))‘1)) + 1) ∈ (0..^𝐾))
268	80, 267	eqeltrd 2890	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 1) ∈ (0..^𝐾))
269	268	adantlr 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 1) ∈ (0..^𝐾))
270	87	ffvelrnda 6828	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ (0..^𝐾))
271		elfzonn0 13077	. . . . . . . . . . . . . . 15 ⊢ (((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ (0..^𝐾) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ ℕ₀)
272	270, 271	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ ℕ₀)
273	272	nn0cnd 11945	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((1^st ‘(1^st ‘𝑇))‘𝑛) ∈ ℂ)
274	273	addid1d 10829	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 0) = ((1^st ‘(1^st ‘𝑇))‘𝑛))
275	274, 270	eqeltrd 2890	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 0) ∈ (0..^𝐾))
276	275	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + 0) ∈ (0..^𝐾))
277	75, 77, 269, 276	ifbothda 4462	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0)) ∈ (0..^𝐾))
278	277	fmpttd 6856	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))):(1...𝑁)⟶(0..^𝐾))
279		ovex 7168	. . . . . . . . 9 ⊢ (0..^𝐾) ∈ V
280	279, 48	elmap 8418	. . . . . . . 8 ⊢ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∈ ((0..^𝐾) ↑_m (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))):(1...𝑁)⟶(0..^𝐾))
281	278, 280	sylibr 237	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∈ ((0..^𝐾) ↑_m (1...𝑁)))
282		simpr 488	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...(𝑁 − 1)))
283		1z 12000	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℤ
284	13, 283	jctil 523	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
285		elfzelz 12902	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℤ)
286	285, 283	jctir 524	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
287		fzaddel 12936	. . . . . . . . . . . . . . . 16 ⊢ (((1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
288	284, 286, 287	syl2an 598	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
289	282, 288	mpbid 235	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))
290	158	oveq2d 7151	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
291	290	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
292	289, 291	eleqtrd 2892	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...𝑁))
293	292	ralrimiva 3149	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁))
294		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → 𝑦 ∈ ((1 + 1)...𝑁))
295		peano2z 12011	. . . . . . . . . . . . . . . . . . 19 ⊢ (1 ∈ ℤ → (1 + 1) ∈ ℤ)
296	283, 295	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (1 + 1) ∈ ℤ
297	11, 296	jctil 523	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
298		elfzelz 12902	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℤ)
299	298, 283	jctir 524	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
300		fzsubel 12938	. . . . . . . . . . . . . . . . 17 ⊢ ((((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
301	297, 299, 300	syl2an 598	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
302	294, 301	mpbid 235	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1)))
303		ax-1cn 10584	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℂ
304	303, 303	pncan3oi 10891	. . . . . . . . . . . . . . . 16 ⊢ ((1 + 1) − 1) = 1
305	304	oveq1i 7145	. . . . . . . . . . . . . . 15 ⊢ (((1 + 1) − 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
306	302, 305	eleqtrdi 2900	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (1...(𝑁 − 1)))
307	298	zcnd 12076	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℂ)
308		elfznn 12931	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℕ)
309	308	nncnd 11641	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℂ)
310		subadd2 10879	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
311	303, 310	mp3an2 1446	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
312	311	bicomd 226	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑛 + 1) = 𝑦 ↔ (𝑦 − 1) = 𝑛))
313		eqcom 2805	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑛 + 1) ↔ (𝑛 + 1) = 𝑦)
314		eqcom 2805	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = (𝑦 − 1) ↔ (𝑦 − 1) = 𝑛)
315	312, 313, 314	3bitr4g 317	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
316	307, 309, 315	syl2an 598	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ((1 + 1)...𝑁) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
317	316	ralrimiva 3149	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ((1 + 1)...𝑁) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
318	317	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
319		reu6i 3667	. . . . . . . . . . . . . 14 ⊢ (((𝑦 − 1) ∈ (1...(𝑁 − 1)) ∧ ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
320	306, 318, 319	syl2anc 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ((1 + 1)...𝑁)) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
321	320	ralrimiva 3149	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑦 ∈ ((1 + 1)...𝑁)∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
322		eqid 2798	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1))
323	322	f1ompt 6852	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁) ↔ (∀𝑛 ∈ (1...(𝑁 − 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁) ∧ ∀𝑦 ∈ ((1 + 1)...𝑁)∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1)))
324	293, 321, 323	sylanbrc 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁))
325		f1osng 6630	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 1 ∈ V) → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
326	1, 128, 325	sylancl 589	. . . . . . . . . . 11 ⊢ (𝜑 → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
327	14, 16	ltnled 10776	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
328	20, 327	mpbid 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
329		elfzle2 12906	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
330	328, 329	nsyl 142	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
331		disjsn 4607	. . . . . . . . . . . 12 ⊢ (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
332	330, 331	sylibr 237	. . . . . . . . . . 11 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)
333		1re 10630	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
334	333	ltp1i 11533	. . . . . . . . . . . . . . 15 ⊢ 1 < (1 + 1)
335	296	zrei 11975	. . . . . . . . . . . . . . . 16 ⊢ (1 + 1) ∈ ℝ
336	333, 335	ltnlei 10750	. . . . . . . . . . . . . . 15 ⊢ (1 < (1 + 1) ↔ ¬ (1 + 1) ≤ 1)
337	334, 336	mpbi 233	. . . . . . . . . . . . . 14 ⊢ ¬ (1 + 1) ≤ 1
338		elfzle1 12905	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ((1 + 1)...𝑁) → (1 + 1) ≤ 1)
339	337, 338	mto 200	. . . . . . . . . . . . 13 ⊢ ¬ 1 ∈ ((1 + 1)...𝑁)
340		disjsn 4607	. . . . . . . . . . . . 13 ⊢ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁))
341	339, 340	mpbir 234	. . . . . . . . . . . 12 ⊢ (((1 + 1)...𝑁) ∩ {1}) = ∅
342	341	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (((1 + 1)...𝑁) ∩ {1}) = ∅)
343		f1oun 6609	. . . . . . . . . . 11 ⊢ ((((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁) ∧ {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1}) ∧ (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ∧ (((1 + 1)...𝑁) ∩ {1}) = ∅)) → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1}))
344	324, 326, 332, 342, 343	syl22anc 837	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1}))
345		elex 3459	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ V)
346	1, 345	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ V)
347	128	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ V)
348	158, 97	eqeltrd 2890	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘1))
349		uzid 12246	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)))
350		peano2uz 12289	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
351	13, 349, 350	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
352	158, 351	eqeltrrd 2891	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
353		fzsplit2 12927	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
354	348, 352, 353	syl2anc 587	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
355	158	oveq1d 7150	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
356		fzsn 12944	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
357	11, 356	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
358	355, 357	eqtrd 2833	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
359	358	uneq2d 4090	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
360	354, 359	eqtr2d 2834	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
361		iftrue 4431	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
362	361	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 = 𝑁) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
363	346, 347, 360, 362	fmptapd 6910	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {⟨𝑁, 1⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
364		eleq1 2877	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑁 → (𝑛 ∈ (1...(𝑁 − 1)) ↔ 𝑁 ∈ (1...(𝑁 − 1))))
365	364	notbid 321	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1))))
366	330, 365	syl5ibrcom 250	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...(𝑁 − 1))))
367	366	necon2ad 3002	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ≠ 𝑁))
368	367	imp 410	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ≠ 𝑁)
369		ifnefalse 4437	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ≠ 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
370	368, 369	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...(𝑁 − 1))) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
371	370	mpteq2dva 5125	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)))
372	371	uneq1d 4089	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {⟨𝑁, 1⟩}) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
373	363, 372	eqtr3d 2835	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
374	354, 359	eqtrd 2833	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
375		uzid 12246	. . . . . . . . . . . . . 14 ⊢ (1 ∈ ℤ → 1 ∈ (ℤ_≥‘1))
376		peano2uz 12289	. . . . . . . . . . . . . 14 ⊢ (1 ∈ (ℤ_≥‘1) → (1 + 1) ∈ (ℤ_≥‘1))
377	283, 375, 376	mp2b 10	. . . . . . . . . . . . 13 ⊢ (1 + 1) ∈ (ℤ_≥‘1)
378		fzsplit2 12927	. . . . . . . . . . . . 13 ⊢ (((1 + 1) ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘1)) → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
379	377, 97, 378	sylancr 590	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
380		fzsn 12944	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ ℤ → (1...1) = {1})
381	283, 380	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (1...1) = {1}
382	381	uneq1i 4086	. . . . . . . . . . . . 13 ⊢ ((1...1) ∪ ((1 + 1)...𝑁)) = ({1} ∪ ((1 + 1)...𝑁))
383	382	equncomi 4082	. . . . . . . . . . . 12 ⊢ ((1...1) ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
384	379, 383	eqtrdi 2849	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1}))
385	373, 374, 384	f1oeq123d 6585	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1})))
386	344, 385	mpbird 260	. . . . . . . . 9 ⊢ (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁))
387		f1oco 6612	. . . . . . . . 9 ⊢ (((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ∧ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁)) → ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
388	93, 386, 387	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
389		f1oeq1 6579	. . . . . . . . 9 ⊢ (𝑓 = ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁)))
390	52, 389	elab 3615	. . . . . . . 8 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
391	388, 390	sylibr 237	. . . . . . 7 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
392		opelxpi 5556	. . . . . . 7 ⊢ (((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∈ ((0..^𝐾) ↑_m (1...𝑁)) ∧ ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩ ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
393	281, 391, 392	syl2anc 587	. . . . . 6 ⊢ (𝜑 → ⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩ ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
394	1	nnnn0d 11943	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
395		nn0fz0 13000	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ ↔ 𝑁 ∈ (0...𝑁))
396	394, 395	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
397		opelxpi 5556	. . . . . 6 ⊢ ((⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩ ∈ (((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 𝑁 ∈ (0...𝑁)) → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
398	393, 396, 397	syl2anc 587	. . . . 5 ⊢ (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
399		elrab3t 3627	. . . . 5 ⊢ ((∀𝑡(𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))) ∧ ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁))) → (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ {𝑡 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))))
400	73, 398, 399	syl2anc 587	. . . 4 ⊢ (𝜑 → (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ {𝑡 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))) ∘_f + (((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))))))
401	7, 400	mpbird 260	. . 3 ⊢ (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ {𝑡 ∈ ((((0..^𝐾) ↑_m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2^nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1^st ‘(1^st ‘𝑡)) ∘_f + ((((2^nd ‘(1^st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2^nd ‘(1^st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))})
402	401, 2	eleqtrrdi 2901	. 2 ⊢ (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ 𝑆)
403		fveqeq2 6654	. . . . . 6 ⊢ (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ = 𝑇 → ((2^nd ‘⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩) = 𝑁 ↔ (2^nd ‘𝑇) = 𝑁))
404	27, 403	syl5ibcom 248	. . . . 5 ⊢ (𝜑 → (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ = 𝑇 → (2^nd ‘𝑇) = 𝑁))
405	1	nnne0d 11675	. . . . . 6 ⊢ (𝜑 → 𝑁 ≠ 0)
406		neeq1 3049	. . . . . 6 ⊢ ((2^nd ‘𝑇) = 𝑁 → ((2^nd ‘𝑇) ≠ 0 ↔ 𝑁 ≠ 0))
407	405, 406	syl5ibrcom 250	. . . . 5 ⊢ (𝜑 → ((2^nd ‘𝑇) = 𝑁 → (2^nd ‘𝑇) ≠ 0))
408	404, 407	syld 47	. . . 4 ⊢ (𝜑 → (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ = 𝑇 → (2^nd ‘𝑇) ≠ 0))
409	408	necon2d 3010	. . 3 ⊢ (𝜑 → ((2^nd ‘𝑇) = 0 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ≠ 𝑇))
410	6, 409	mpd 15	. 2 ⊢ (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ≠ 𝑇)
411		neeq1 3049	. . 3 ⊢ (𝑧 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ → (𝑧 ≠ 𝑇 ↔ ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ≠ 𝑇))
412	411	rspcev 3571	. 2 ⊢ ((⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ∈ 𝑆 ∧ ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1^st ‘(1^st ‘𝑇))‘𝑛) + if(𝑛 = ((2^nd ‘(1^st ‘𝑇))‘1), 1, 0))), ((2^nd ‘(1^st ‘𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))⟩, 𝑁⟩ ≠ 𝑇) → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
413	402, 410, 412	syl2anc 587	1 ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 ∀wal 1536 = wceq 1538 ∈ wcel 2111 {cab 2776 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ∃!wreu 3108 {crab 3110 Vcvv 3441 ⦋csb 3828 ∪ cun 3879 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 ifcif 4425 {csn 4525 ⟨cop 4531 class class class wbr 5030 ↦ cmpt 5110 × cxp 5517 ^◡ccnv 5518 ran crn 5520 “ cima 5522 ∘ ccom 5523 Fun wfun 6318 Fn wfn 6319 ⟶wf 6320 –onto→wfo 6322 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ∘_f cof 7387 1^st c1st 7669 2^nd c2nd 7670 ↑_m cmap 8389 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 < clt 10664 ≤ cle 10665 − cmin 10859 ℕcn 11625 ℕ₀cn0 11885 ℤcz 11969 ℤ_≥cuz 12231 ...cfz 12885 ..^cfzo 13028

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-fzo 13029

This theorem is referenced by: poimirlem18 35075

W3C validator