Theorem poimirlem21 37635

Description: Lemma for poimir 37647 stating that, given a face not on a back face of the cube and a simplex in which it's opposite the final point of the walk, there exists exactly one other simplex containing it. (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 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem22.1	⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
poimirlem22.2	⊢ (𝜑 → 𝑇 ∈ 𝑆)
poimirlem22.3	⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0)
poimirlem21.4	⊢ (𝜑 → (2nd ‘𝑇) = 𝑁)

Assertion

Ref	Expression
poimirlem21	⊢ (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 poimirlem21

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		poimir.0	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		poimirlem22.s	. . 3 ⊢ 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 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
3		poimirlem22.1	. . 3 ⊢ (𝜑 → 𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑m (1...𝑁)))
4		poimirlem22.2	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
5		poimirlem22.3	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘𝑛) ≠ 0)
6		poimirlem21.4	. . 3 ⊢ (𝜑 → (2nd ‘𝑇) = 𝑁)
7	1, 2, 3, 4, 5, 6	poimirlem20 37634	. 2 ⊢ (𝜑 → ∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
8	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (2nd ‘𝑇) = 𝑁)
9	1	nnred 12201	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℝ)
10	9	ltm1d 12115	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 − 1) < 𝑁)
11		nnm1nn0 12483	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
12	1, 11	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ0)
13	12	nn0red 12504	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑁 − 1) ∈ ℝ)
14	13, 9	ltnled 11321	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
15	10, 14	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
16		elfzle2 13489	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
17	15, 16	nsyl 140	. . . . . . . . . . . . 13 ⊢ (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
18		eleq1 2816	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑧) = 𝑁 → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ↔ 𝑁 ∈ (1...(𝑁 − 1))))
19	18	notbid 318	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝑧) = 𝑁 → (¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1))))
20	17, 19	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2nd ‘𝑧) = 𝑁 → ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))
21	20	necon2ad 2940	. . . . . . . . . . 11 ⊢ (𝜑 → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) → (2nd ‘𝑧) ≠ 𝑁))
22	21	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) → (2nd ‘𝑧) ≠ 𝑁))
23	1	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → 𝑁 ∈ ℕ)
24		fveq2 6858	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (2nd ‘𝑡) = (2nd ‘𝑧))
25	24	breq2d 5119	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑧 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑧)))
26	25	ifbid 4512	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑧 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)))
27	26	csbeq1d 3866	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑧 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
28		2fveq3 6863	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑧 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑧)))
29		2fveq3 6863	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑧 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑧)))
30	29	imaeq1d 6030	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑧 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑧)) “ (1...𝑗)))
31	30	xpeq1d 5667	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}))
32	29	imaeq1d 6030	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑧 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)))
33	32	xpeq1d 5667	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑧 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))
34	31, 33	uneq12d 4132	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑧 → ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))
35	28, 34	oveq12d 7405	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑧 → ((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑧)) ∘f + ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))
36	35	csbeq2dv 3869	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = 𝑧 → ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑧)) ∘f + ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))
37	27, 36	eqtrd 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = 𝑧 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑧)) ∘f + ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))
38	37	mpteq2dv 5201	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑧 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑧)) ∘f + ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))))
39	38	eqeq2d 2740	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑧 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑧)) ∘f + ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
40	39, 2	elrab2 3662	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 ↔ (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑧)) ∘f + ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
41	40	simprbi 496	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑧)) ∘f + ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))))
42	41	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑧), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑧)) ∘f + ((((2nd ‘(1st ‘𝑧)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑧)) “ ((𝑗 + 1)...𝑁)) × {0})))))
43		elrabi 3654	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ {𝑡 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
44	43, 2	eleq2s 2846	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ 𝑆 → 𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
45		xp1st 8000	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
46	44, 45	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝑆 → (1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
47		xp1st 8000	. . . . . . . . . . . . . . . . . 18 ⊢ ((1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st ‘𝑧)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
48	46, 47	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ 𝑆 → (1st ‘(1st ‘𝑧)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
49		elmapi 8822	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘(1st ‘𝑧)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶(0..^𝐾))
50	48, 49	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶(0..^𝐾))
51		elfzoelz 13620	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (0..^𝐾) → 𝑛 ∈ ℤ)
52	51	ssriv 3950	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝐾) ⊆ ℤ
53		fss 6704	. . . . . . . . . . . . . . . 16 ⊢ (((1st ‘(1st ‘𝑧)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶ℤ)
54	50, 52, 53	sylancl 586	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶ℤ)
55	54	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (1st ‘(1st ‘𝑧)):(1...𝑁)⟶ℤ)
56		xp2nd 8001	. . . . . . . . . . . . . . . . 17 ⊢ ((1st ‘𝑧) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
57	46, 56	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
58		fvex 6871	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘(1st ‘𝑧)) ∈ V
59		f1oeq1 6788	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = (2nd ‘(1st ‘𝑧)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁)))
60	58, 59	elab 3646	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘(1st ‘𝑧)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
61	57, 60	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
62	61	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘(1st ‘𝑧)):(1...𝑁)–1-1-onto→(1...𝑁))
63		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))
64	23, 42, 55, 62, 63	poimirlem1 37615	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛))
65	1	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → 𝑁 ∈ ℕ)
66		fveq2 6858	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑇 → (2nd ‘𝑡) = (2nd ‘𝑇))
67	66	breq2d 5119	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑇 → (𝑦 < (2nd ‘𝑡) ↔ 𝑦 < (2nd ‘𝑇)))
68	67	ifbid 4512	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑇 → if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)))
69	68	csbeq1d 3866	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
70		2fveq3 6863	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑇 → (1st ‘(1st ‘𝑡)) = (1st ‘(1st ‘𝑇)))
71		2fveq3 6863	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑡 = 𝑇 → (2nd ‘(1st ‘𝑡)) = (2nd ‘(1st ‘𝑇)))
72	71	imaeq1d 6030	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) = ((2nd ‘(1st ‘𝑇)) “ (1...𝑗)))
73	72	xpeq1d 5667	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}))
74	71	imaeq1d 6030	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑡 = 𝑇 → ((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)))
75	74	xpeq1d 5667	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑡 = 𝑇 → (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
76	73, 75	uneq12d 4132	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑡 = 𝑇 → ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
77	70, 76	oveq12d 7405	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑡 = 𝑇 → ((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st ‘𝑇)) ∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
78	77	csbeq2dv 3869	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
79	69, 78	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑡 = 𝑇 → ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
80	79	mpteq2dv 5201	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
81	80	eqeq2d 2740	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
82	81, 2	elrab2 3662	. . . . . . . . . . . . . . . . . . . 20 ⊢ (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 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
83	82	simprbi 496	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑇 ∈ 𝑆 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
84	4, 83	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
85	84	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < (2nd ‘𝑇), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑇)) ∘f + ((((2nd ‘(1st ‘𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
86		elrabi 3654	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (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 < (2nd ‘𝑡), 𝑦, (𝑦 + 1)) / 𝑗⦌((1st ‘(1st ‘𝑡)) ∘f + ((((2nd ‘(1st ‘𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st ‘𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
87	86, 2	eleq2s 2846	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
88	4, 87	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
89		xp1st 8000	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
90	88, 89	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
91		xp1st 8000	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
92	90, 91	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
93		elmapi 8822	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘(1st ‘𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
94	92, 93	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾))
95		fss 6704	. . . . . . . . . . . . . . . . . . 19 ⊢ (((1st ‘(1st ‘𝑇)):(1...𝑁)⟶(0..^𝐾) ∧ (0..^𝐾) ⊆ ℤ) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
96	94, 52, 95	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
97	96	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (1st ‘(1st ‘𝑇)):(1...𝑁)⟶ℤ)
98		xp2nd 8001	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1st ‘𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
99	90, 98	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
100		fvex 6871	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2nd ‘(1st ‘𝑇)) ∈ V
101		f1oeq1 6788	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑓 = (2nd ‘(1st ‘𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
102	100, 101	elab 3646	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘(1st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
103	99, 102	sylib 218	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
104	103	ad2antrr 726	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘(1st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
105		simplr 768	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))
106		xp2nd 8001	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑇) ∈ (0...𝑁))
107	88, 106	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (2nd ‘𝑇) ∈ (0...𝑁))
108	107	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑇) ∈ (0...𝑁))
109		eldifsn 4750	. . . . . . . . . . . . . . . . . . 19 ⊢ ((2nd ‘𝑇) ∈ ((0...𝑁) ∖ {(2nd ‘𝑧)}) ↔ ((2nd ‘𝑇) ∈ (0...𝑁) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)))
110	109	biimpri 228	. . . . . . . . . . . . . . . . . 18 ⊢ (((2nd ‘𝑇) ∈ (0...𝑁) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘𝑇) ∈ ((0...𝑁) ∖ {(2nd ‘𝑧)}))
111	108, 110	sylan 580	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → (2nd ‘𝑇) ∈ ((0...𝑁) ∖ {(2nd ‘𝑧)}))
112	65, 85, 97, 104, 105, 111	poimirlem2 37616	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ∧ (2nd ‘𝑇) ≠ (2nd ‘𝑧)) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛))
113	112	ex 412	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → ((2nd ‘𝑇) ≠ (2nd ‘𝑧) → ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛)))
114	113	necon1bd 2943	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛) → (2nd ‘𝑇) = (2nd ‘𝑧)))
115	114	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘((2nd ‘𝑧) − 1))‘𝑛) ≠ ((𝐹‘(2nd ‘𝑧))‘𝑛) → (2nd ‘𝑇) = (2nd ‘𝑧)))
116	64, 115	mpd 15	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑇) = (2nd ‘𝑧))
117	116	neeq1d 2984	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝑆) ∧ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → ((2nd ‘𝑇) ≠ 𝑁 ↔ (2nd ‘𝑧) ≠ 𝑁))
118	117	exbiri 810	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) → ((2nd ‘𝑧) ≠ 𝑁 → (2nd ‘𝑇) ≠ 𝑁)))
119	22, 118	mpdd 43	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑧) ∈ (1...(𝑁 − 1)) → (2nd ‘𝑇) ≠ 𝑁))
120	119	necon2bd 2941	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑇) = 𝑁 → ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))
121	8, 120	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))
122		xp2nd 8001	. . . . . . . . 9 ⊢ (𝑧 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (2nd ‘𝑧) ∈ (0...𝑁))
123	44, 122	syl 17	. . . . . . . 8 ⊢ (𝑧 ∈ 𝑆 → (2nd ‘𝑧) ∈ (0...𝑁))
124		nn0uz 12835	. . . . . . . . . . . . . . . . . 18 ⊢ ℕ0 = (ℤ≥‘0)
125	12, 124	eleqtrdi 2838	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 − 1) ∈ (ℤ≥‘0))
126		fzpred 13533	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘0) → (0...(𝑁 − 1)) = ({0} ∪ ((0 + 1)...(𝑁 − 1))))
127	125, 126	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0...(𝑁 − 1)) = ({0} ∪ ((0 + 1)...(𝑁 − 1))))
128		0p1e1 12303	. . . . . . . . . . . . . . . . . 18 ⊢ (0 + 1) = 1
129	128	oveq1i 7397	. . . . . . . . . . . . . . . . 17 ⊢ ((0 + 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
130	129	uneq2i 4128	. . . . . . . . . . . . . . . 16 ⊢ ({0} ∪ ((0 + 1)...(𝑁 − 1))) = ({0} ∪ (1...(𝑁 − 1)))
131	127, 130	eqtrdi 2780	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0...(𝑁 − 1)) = ({0} ∪ (1...(𝑁 − 1))))
132	131	eleq2d 2814	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2nd ‘𝑧) ∈ (0...(𝑁 − 1)) ↔ (2nd ‘𝑧) ∈ ({0} ∪ (1...(𝑁 − 1)))))
133	132	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝜑 → (¬ (2nd ‘𝑧) ∈ (0...(𝑁 − 1)) ↔ ¬ (2nd ‘𝑧) ∈ ({0} ∪ (1...(𝑁 − 1)))))
134		ioran 985	. . . . . . . . . . . . . 14 ⊢ (¬ ((2nd ‘𝑧) = 0 ∨ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ↔ (¬ (2nd ‘𝑧) = 0 ∧ ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))
135		elun 4116	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑧) ∈ ({0} ∪ (1...(𝑁 − 1))) ↔ ((2nd ‘𝑧) ∈ {0} ∨ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))
136		fvex 6871	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘𝑧) ∈ V
137	136	elsn 4604	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑧) ∈ {0} ↔ (2nd ‘𝑧) = 0)
138	137	orbi1i 913	. . . . . . . . . . . . . . 15 ⊢ (((2nd ‘𝑧) ∈ {0} ∨ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) ↔ ((2nd ‘𝑧) = 0 ∨ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))
139	135, 138	bitri 275	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑧) ∈ ({0} ∪ (1...(𝑁 − 1))) ↔ ((2nd ‘𝑧) = 0 ∨ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))
140	134, 139	xchnxbir 333	. . . . . . . . . . . . 13 ⊢ (¬ (2nd ‘𝑧) ∈ ({0} ∪ (1...(𝑁 − 1))) ↔ (¬ (2nd ‘𝑧) = 0 ∧ ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))
141	133, 140	bitrdi 287	. . . . . . . . . . . 12 ⊢ (𝜑 → (¬ (2nd ‘𝑧) ∈ (0...(𝑁 − 1)) ↔ (¬ (2nd ‘𝑧) = 0 ∧ ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))))
142	141	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ ¬ (2nd ‘𝑧) ∈ (0...(𝑁 − 1))) ↔ ((2nd ‘𝑧) ∈ (0...𝑁) ∧ (¬ (2nd ‘𝑧) = 0 ∧ ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))))))
143		uncom 4121	. . . . . . . . . . . . . . . 16 ⊢ ((0...(𝑁 − 1)) ∪ {𝑁}) = ({𝑁} ∪ (0...(𝑁 − 1)))
144	143	difeq1i 4085	. . . . . . . . . . . . . . 15 ⊢ (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ (0...(𝑁 − 1))) = (({𝑁} ∪ (0...(𝑁 − 1))) ∖ (0...(𝑁 − 1)))
145		difun2 4444	. . . . . . . . . . . . . . 15 ⊢ (({𝑁} ∪ (0...(𝑁 − 1))) ∖ (0...(𝑁 − 1))) = ({𝑁} ∖ (0...(𝑁 − 1)))
146	144, 145	eqtri 2752	. . . . . . . . . . . . . 14 ⊢ (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ (0...(𝑁 − 1))) = ({𝑁} ∖ (0...(𝑁 − 1)))
147	1	nncnd 12202	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℂ)
148		npcan1 11603	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
149	147, 148	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
150	1	nnnn0d 12503	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
151	150, 124	eleqtrdi 2838	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘0))
152	149, 151	eqeltrd 2828	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘0))
153	12	nn0zd 12555	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (𝑁 − 1) ∈ ℤ)
154		uzid 12808	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)))
155		peano2uz 12860	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 − 1) ∈ (ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
156	153, 154, 155	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ≥‘(𝑁 − 1)))
157	149, 156	eqeltrrd 2829	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1)))
158		fzsplit2 13510	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 − 1) + 1) ∈ (ℤ≥‘0) ∧ 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) → (0...𝑁) = ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
159	152, 157, 158	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
160	149	oveq1d 7402	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
161	1	nnzd 12556	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑁 ∈ ℤ)
162		fzsn 13527	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
163	161, 162	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑁...𝑁) = {𝑁})
164	160, 163	eqtrd 2764	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
165	164	uneq2d 4131	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((0...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((0...(𝑁 − 1)) ∪ {𝑁}))
166	159, 165	eqtrd 2764	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0...𝑁) = ((0...(𝑁 − 1)) ∪ {𝑁}))
167	166	difeq1d 4088	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((0...𝑁) ∖ (0...(𝑁 − 1))) = (((0...(𝑁 − 1)) ∪ {𝑁}) ∖ (0...(𝑁 − 1))))
168		elfzle2 13489	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ (0...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
169	15, 168	nsyl 140	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ¬ 𝑁 ∈ (0...(𝑁 − 1)))
170		incom 4172	. . . . . . . . . . . . . . . . 17 ⊢ ((0...(𝑁 − 1)) ∩ {𝑁}) = ({𝑁} ∩ (0...(𝑁 − 1)))
171	170	eqeq1i 2734	. . . . . . . . . . . . . . . 16 ⊢ (((0...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ({𝑁} ∩ (0...(𝑁 − 1))) = ∅)
172		disjsn 4675	. . . . . . . . . . . . . . . 16 ⊢ (((0...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (0...(𝑁 − 1)))
173		disj3 4417	. . . . . . . . . . . . . . . 16 ⊢ (({𝑁} ∩ (0...(𝑁 − 1))) = ∅ ↔ {𝑁} = ({𝑁} ∖ (0...(𝑁 − 1))))
174	171, 172, 173	3bitr3i 301	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑁 ∈ (0...(𝑁 − 1)) ↔ {𝑁} = ({𝑁} ∖ (0...(𝑁 − 1))))
175	169, 174	sylib 218	. . . . . . . . . . . . . 14 ⊢ (𝜑 → {𝑁} = ({𝑁} ∖ (0...(𝑁 − 1))))
176	146, 167, 175	3eqtr4a 2790	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((0...𝑁) ∖ (0...(𝑁 − 1))) = {𝑁})
177	176	eleq2d 2814	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2nd ‘𝑧) ∈ ((0...𝑁) ∖ (0...(𝑁 − 1))) ↔ (2nd ‘𝑧) ∈ {𝑁}))
178		eldif 3924	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑧) ∈ ((0...𝑁) ∖ (0...(𝑁 − 1))) ↔ ((2nd ‘𝑧) ∈ (0...𝑁) ∧ ¬ (2nd ‘𝑧) ∈ (0...(𝑁 − 1))))
179	136	elsn 4604	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑧) ∈ {𝑁} ↔ (2nd ‘𝑧) = 𝑁)
180	177, 178, 179	3bitr3g 313	. . . . . . . . . . 11 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ ¬ (2nd ‘𝑧) ∈ (0...(𝑁 − 1))) ↔ (2nd ‘𝑧) = 𝑁))
181	142, 180	bitr3d 281	. . . . . . . . . 10 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ (¬ (2nd ‘𝑧) = 0 ∧ ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))) ↔ (2nd ‘𝑧) = 𝑁))
182	181	biimpd 229	. . . . . . . . 9 ⊢ (𝜑 → (((2nd ‘𝑧) ∈ (0...𝑁) ∧ (¬ (2nd ‘𝑧) = 0 ∧ ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1)))) → (2nd ‘𝑧) = 𝑁))
183	182	expdimp 452	. . . . . . . 8 ⊢ ((𝜑 ∧ (2nd ‘𝑧) ∈ (0...𝑁)) → ((¬ (2nd ‘𝑧) = 0 ∧ ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑧) = 𝑁))
184	123, 183	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((¬ (2nd ‘𝑧) = 0 ∧ ¬ (2nd ‘𝑧) ∈ (1...(𝑁 − 1))) → (2nd ‘𝑧) = 𝑁))
185	121, 184	mpan2d 694	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (¬ (2nd ‘𝑧) = 0 → (2nd ‘𝑧) = 𝑁))
186	1, 2, 3	poimirlem14 37628	. . . . . . . . . 10 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 𝑁)
187		fveqeq2 6867	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑠 → ((2nd ‘𝑧) = 𝑁 ↔ (2nd ‘𝑠) = 𝑁))
188	187	rmo4 3701	. . . . . . . . . 10 ⊢ (∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 𝑁 ↔ ∀𝑧 ∈ 𝑆 ∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑠) = 𝑁) → 𝑧 = 𝑠))
189	186, 188	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 ∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑠) = 𝑁) → 𝑧 = 𝑠))
190	189	r19.21bi 3229	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑠) = 𝑁) → 𝑧 = 𝑠))
191	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → 𝑇 ∈ 𝑆)
192		fveqeq2 6867	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑇 → ((2nd ‘𝑠) = 𝑁 ↔ (2nd ‘𝑇) = 𝑁))
193	192	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑠) = 𝑁) ↔ ((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑇) = 𝑁)))
194		eqeq2 2741	. . . . . . . . . 10 ⊢ (𝑠 = 𝑇 → (𝑧 = 𝑠 ↔ 𝑧 = 𝑇))
195	193, 194	imbi12d 344	. . . . . . . . 9 ⊢ (𝑠 = 𝑇 → ((((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑠) = 𝑁) → 𝑧 = 𝑠) ↔ (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑇) = 𝑁) → 𝑧 = 𝑇)))
196	195	rspccv 3585	. . . . . . . 8 ⊢ (∀𝑠 ∈ 𝑆 (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑠) = 𝑁) → 𝑧 = 𝑠) → (𝑇 ∈ 𝑆 → (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑇) = 𝑁) → 𝑧 = 𝑇)))
197	190, 191, 196	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (((2nd ‘𝑧) = 𝑁 ∧ (2nd ‘𝑇) = 𝑁) → 𝑧 = 𝑇))
198	8, 197	mpan2d 694	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → ((2nd ‘𝑧) = 𝑁 → 𝑧 = 𝑇))
199	185, 198	syld 47	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (¬ (2nd ‘𝑧) = 0 → 𝑧 = 𝑇))
200	199	necon1ad 2942	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝑧 ≠ 𝑇 → (2nd ‘𝑧) = 0))
201	200	ralrimiva 3125	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝑧 ≠ 𝑇 → (2nd ‘𝑧) = 0))
202	1, 2, 3	poimirlem13 37627	. . 3 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 0)
203		rmoim 3711	. . 3 ⊢ (∀𝑧 ∈ 𝑆 (𝑧 ≠ 𝑇 → (2nd ‘𝑧) = 0) → (∃*𝑧 ∈ 𝑆 (2nd ‘𝑧) = 0 → ∃*𝑧 ∈ 𝑆 𝑧 ≠ 𝑇))
204	201, 202, 203	sylc 65	. 2 ⊢ (𝜑 → ∃*𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)
205		reu5 3356	. 2 ⊢ (∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇 ↔ (∃𝑧 ∈ 𝑆 𝑧 ≠ 𝑇 ∧ ∃*𝑧 ∈ 𝑆 𝑧 ≠ 𝑇))
206	7, 204, 205	sylanbrc 583	1 ⊢ (𝜑 → ∃!𝑧 ∈ 𝑆 𝑧 ≠ 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  {cab 2707   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  ∃!wreu 3352  ∃*wrmo 3353  {crab 3405  ⦋csb 3862   ∖ cdif 3911   ∪ cun 3912   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  ifcif 4488  {csn 4589   class class class wbr 5107   ↦ cmpt 5188   × cxp 5636  ran crn 5639   “ cima 5641  ⟶wf 6507  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651  1^st c1st 7966  2^nd c2nd 7967   ↑_m cmap 8799  ℂcc 11066  0cc0 11068  1c1 11069   + caddc 11071   < clt 11208   ≤ cle 11209   − cmin 11405  ℕcn 12186  ℕ₀cn0 12442  ℤcz 12529  ℤ_≥cuz 12793  ...cfz 13468  ..^cfzo 13615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469  df-fzo 13616

This theorem is referenced by:  poimirlem22  37636