poimirlem9 - Mathbox for Brendan Leahy

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Theorem poimirlem9 33749

Description: Lemma for poimir 33773, establishing the two walks that yield a given face when the opposite vertex is neither first nor last. (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}))))}
poimirlem9.1	⊢ (𝜑 → 𝑇 ∈ 𝑆)
poimirlem9.2	⊢ (𝜑 → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
poimirlem9.3	⊢ (𝜑 → 𝑈 ∈ 𝑆)
poimirlem9.4	⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)) ≠ (2^nd ‘(1^st ‘𝑇)))

**Assertion**
Ref	Expression
poimirlem9	⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))))

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

**Proof of Theorem poimirlem9**
Step	Hyp	Ref	Expression
1		resundi 5627	. . . 4 ⊢ ((2^nd ‘(1^st ‘𝑈)) ↾ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ ((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
2		poimir.0	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
3	2	nncnd 11332	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
4		npcan1 10749	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
5	3, 4	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
6	2	nnzd 11766	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
7		peano2zm 11705	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
8		uzid 11938	. . . . . . . . . . . 12 ⊢ ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)))
9		peano2uz 11978	. . . . . . . . . . . 12 ⊢ ((𝑁 − 1) ∈ (ℤ_≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
10	6, 7, 8, 9	4syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ_≥‘(𝑁 − 1)))
11	5, 10	eqeltrrd 2897	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘(𝑁 − 1)))
12		fzss2 12623	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
13	11, 12	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
14		poimirlem9.2	. . . . . . . . 9 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ (1...(𝑁 − 1)))
15	13, 14	sseldd 3810	. . . . . . . 8 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ (1...𝑁))
16		fzp1elp1 12636	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → ((2^nd ‘𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
17	14, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ (1...((𝑁 − 1) + 1)))
18	5	oveq2d 6899	. . . . . . . . 9 ⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁))
19	17, 18	eleqtrd 2898	. . . . . . . 8 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ∈ (1...𝑁))
20	15, 19	prssd 4554	. . . . . . 7 ⊢ (𝜑 → {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ⊆ (1...𝑁))
21		undif 4256	. . . . . . 7 ⊢ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ⊆ (1...𝑁) ↔ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = (1...𝑁))
22	20, 21	sylib 209	. . . . . 6 ⊢ (𝜑 → ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = (1...𝑁))
23	22	reseq2d 5610	. . . . 5 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) ↾ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = ((2^nd ‘(1^st ‘𝑈)) ↾ (1...𝑁)))
24		poimirlem9.3	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
25		elrabi 3565	. . . . . . . . 9 ⊢ (𝑈 ∈ {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}))))} → 𝑈 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
26		poimirlem22.s	. . . . . . . . 9 ⊢ 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}))))}
27	25, 26	eleq2s 2914	. . . . . . . 8 ⊢ (𝑈 ∈ 𝑆 → 𝑈 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
28		xp1st 7439	. . . . . . . 8 ⊢ (𝑈 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1^st ‘𝑈) ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
29		xp2nd 7440	. . . . . . . 8 ⊢ ((1^st ‘𝑈) ∈ (((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2^nd ‘(1^st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
30	24, 27, 28, 29	4syl 19	. . . . . . 7 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
31		fvex 6430	. . . . . . . 8 ⊢ (2^nd ‘(1^st ‘𝑈)) ∈ V
32		f1oeq1 6352	. . . . . . . 8 ⊢ (𝑓 = (2^nd ‘(1^st ‘𝑈)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁)))
33	31, 32	elab 3556	. . . . . . 7 ⊢ ((2^nd ‘(1^st ‘𝑈)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
34	30, 33	sylib 209	. . . . . 6 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁))
35		f1ofn 6363	. . . . . 6 ⊢ ((2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑈)) Fn (1...𝑁))
36		fnresdm 6220	. . . . . 6 ⊢ ((2^nd ‘(1^st ‘𝑈)) Fn (1...𝑁) → ((2^nd ‘(1^st ‘𝑈)) ↾ (1...𝑁)) = (2^nd ‘(1^st ‘𝑈)))
37	34, 35, 36	3syl 18	. . . . 5 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) ↾ (1...𝑁)) = (2^nd ‘(1^st ‘𝑈)))
38	23, 37	eqtrd 2851	. . . 4 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) ↾ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (2^nd ‘(1^st ‘𝑈)))
39	1, 38	syl5eqr 2865	. . 3 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ ((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (2^nd ‘(1^st ‘𝑈)))
40		2lt3 11490	. . . . . 6 ⊢ 2 < 3
41		2re 11386	. . . . . . 7 ⊢ 2 ∈ ℝ
42		3re 11392	. . . . . . 7 ⊢ 3 ∈ ℝ
43	41, 42	ltnlei 10452	. . . . . 6 ⊢ (2 < 3 ↔ ¬ 3 ≤ 2)
44	40, 43	mpbi 221	. . . . 5 ⊢ ¬ 3 ≤ 2
45		df-pr 4384	. . . . . . . . . . . 12 ⊢ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} = ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩} ∪ {⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})
46	45	coeq2i 5497	. . . . . . . . . . 11 ⊢ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩} ∪ {⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}))
47		coundi 5863	. . . . . . . . . . 11 ⊢ ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩} ∪ {⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})) = (((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩}) ∪ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}))
48	46, 47	eqtri 2839	. . . . . . . . . 10 ⊢ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) = (((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩}) ∪ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}))
49		poimirlem9.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
50		elrabi 3565	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (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)))
51	50, 26	eleq2s 2914	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇 ∈ 𝑆 → 𝑇 ∈ ((((0..^𝐾) ↑_𝑚 (1...𝑁)) × {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
52		xp1st 7439	. . . . . . . . . . . . . . . . . . . 20 ⊢ (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)}))
53		xp2nd 7440	. . . . . . . . . . . . . . . . . . . 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)})
54	49, 51, 52, 53	4syl 19	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) ∈ {𝑓 ∣ 𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
55		fvex 6430	. . . . . . . . . . . . . . . . . . . 20 ⊢ (2^nd ‘(1^st ‘𝑇)) ∈ V
56		f1oeq1 6352	. . . . . . . . . . . . . . . . . . . 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)))
57	55, 56	elab 3556	. . . . . . . . . . . . . . . . . . 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))
58	54, 57	sylib 209	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
59		f1of1 6361	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1→(1...𝑁))
60	58, 59	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1→(1...𝑁))
61	19	snssd 4541	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → {((2^nd ‘𝑇) + 1)} ⊆ (1...𝑁))
62		f1ores 6376	. . . . . . . . . . . . . . . . 17 ⊢ (((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1→(1...𝑁) ∧ {((2^nd ‘𝑇) + 1)} ⊆ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1)}):{((2^nd ‘𝑇) + 1)}–1-1-onto→((2^nd ‘(1^st ‘𝑇)) “ {((2^nd ‘𝑇) + 1)}))
63	60, 61, 62	syl2anc 575	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1)}):{((2^nd ‘𝑇) + 1)}–1-1-onto→((2^nd ‘(1^st ‘𝑇)) “ {((2^nd ‘𝑇) + 1)}))
64		f1of 6362	. . . . . . . . . . . . . . . 16 ⊢ (((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1)}):{((2^nd ‘𝑇) + 1)}–1-1-onto→((2^nd ‘(1^st ‘𝑇)) “ {((2^nd ‘𝑇) + 1)}) → ((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1)}):{((2^nd ‘𝑇) + 1)}⟶((2^nd ‘(1^st ‘𝑇)) “ {((2^nd ‘𝑇) + 1)}))
65	63, 64	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1)}):{((2^nd ‘𝑇) + 1)}⟶((2^nd ‘(1^st ‘𝑇)) “ {((2^nd ‘𝑇) + 1)}))
66		f1ofn 6363	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
67	58, 66	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁))
68		fnsnfv 6488	. . . . . . . . . . . . . . . . 17 ⊢ (((2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁) ∧ ((2^nd ‘𝑇) + 1) ∈ (1...𝑁)) → {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))} = ((2^nd ‘(1^st ‘𝑇)) “ {((2^nd ‘𝑇) + 1)}))
69	67, 19, 68	syl2anc 575	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))} = ((2^nd ‘(1^st ‘𝑇)) “ {((2^nd ‘𝑇) + 1)}))
70	69	feq3d 6252	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1)}):{((2^nd ‘𝑇) + 1)}⟶{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))} ↔ ((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1)}):{((2^nd ‘𝑇) + 1)}⟶((2^nd ‘(1^st ‘𝑇)) “ {((2^nd ‘𝑇) + 1)})))
71	65, 70	mpbird 248	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1)}):{((2^nd ‘𝑇) + 1)}⟶{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))})
72		eqid 2817	. . . . . . . . . . . . . . 15 ⊢ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩} = {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩}
73		fvex 6430	. . . . . . . . . . . . . . . 16 ⊢ (2^nd ‘𝑇) ∈ V
74		ovex 6915	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘𝑇) + 1) ∈ V
75	73, 74	fsn 6634	. . . . . . . . . . . . . . 15 ⊢ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩}:{(2^nd ‘𝑇)}⟶{((2^nd ‘𝑇) + 1)} ↔ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩} = {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩})
76	72, 75	mpbir 222	. . . . . . . . . . . . . 14 ⊢ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩}:{(2^nd ‘𝑇)}⟶{((2^nd ‘𝑇) + 1)}
77		fco2 6283	. . . . . . . . . . . . . 14 ⊢ ((((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1)}):{((2^nd ‘𝑇) + 1)}⟶{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))} ∧ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩}:{(2^nd ‘𝑇)}⟶{((2^nd ‘𝑇) + 1)}) → ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩}):{(2^nd ‘𝑇)}⟶{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))})
78	71, 76, 77	sylancl 576	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩}):{(2^nd ‘𝑇)}⟶{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))})
79		fvex 6430	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)) ∈ V
80	79	fconst2 6704	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩}):{(2^nd ‘𝑇)}⟶{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))} ↔ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩}) = ({(2^nd ‘𝑇)} × {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))}))
81	78, 80	sylib 209	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩}) = ({(2^nd ‘𝑇)} × {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))}))
82	73, 79	xpsn 6639	. . . . . . . . . . . 12 ⊢ ({(2^nd ‘𝑇)} × {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))}) = {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}
83	81, 82	syl6eq 2867	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩}) = {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩})
84	83	uneq1d 3976	. . . . . . . . . 10 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩}) ∪ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})) = ({⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ∪ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})))
85	48, 84	syl5eq 2863	. . . . . . . . 9 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) = ({⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ∪ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})))
86		elfznn 12612	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘𝑇) ∈ (1...(𝑁 − 1)) → (2^nd ‘𝑇) ∈ ℕ)
87	14, 86	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℕ)
88	87	nnred 11331	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘𝑇) ∈ ℝ)
89	88	ltp1d 11248	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (2^nd ‘𝑇) < ((2^nd ‘𝑇) + 1))
90	88, 89	ltned 10467	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (2^nd ‘𝑇) ≠ ((2^nd ‘𝑇) + 1))
91	90	necomd 3044	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2^nd ‘𝑇) + 1) ≠ (2^nd ‘𝑇))
92		f1veqaeq 6747	. . . . . . . . . . . . . . 15 ⊢ (((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1→(1...𝑁) ∧ (((2^nd ‘𝑇) + 1) ∈ (1...𝑁) ∧ (2^nd ‘𝑇) ∈ (1...𝑁))) → (((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)) = ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) → ((2^nd ‘𝑇) + 1) = (2^nd ‘𝑇)))
93	60, 19, 15, 92	syl12anc 856	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)) = ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) → ((2^nd ‘𝑇) + 1) = (2^nd ‘𝑇)))
94	93	necon3d 3010	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((2^nd ‘𝑇) + 1) ≠ (2^nd ‘𝑇) → ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)) ≠ ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))))
95	91, 94	mpd 15	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)) ≠ ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)))
96	95	neneqd 2994	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)) = ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)))
97	73, 79	opth 5147	. . . . . . . . . . . 12 ⊢ (⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩ ↔ ((2^nd ‘𝑇) = (2^nd ‘𝑇) ∧ ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)) = ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))))
98	97	simprbi 486	. . . . . . . . . . 11 ⊢ (⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩ → ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)) = ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)))
99	96, 98	nsyl 137	. . . . . . . . . 10 ⊢ (𝜑 → ¬ ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩)
100	90	neneqd 2994	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ (2^nd ‘𝑇) = ((2^nd ‘𝑇) + 1))
101	73, 79	opth1 5146	. . . . . . . . . . 11 ⊢ (⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ → (2^nd ‘𝑇) = ((2^nd ‘𝑇) + 1))
102	100, 101	nsyl 137	. . . . . . . . . 10 ⊢ (𝜑 → ¬ ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩)
103		opex 5135	. . . . . . . . . . . . . . 15 ⊢ ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ ∈ V
104	103	snid 4413	. . . . . . . . . . . . . 14 ⊢ ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ ∈ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}
105		elun1 3990	. . . . . . . . . . . . . 14 ⊢ (⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ ∈ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} → ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ ∈ ({⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ∪ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})))
106	104, 105	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ ∈ ({⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ∪ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}))
107		eleq2 2885	. . . . . . . . . . . . 13 ⊢ (({⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ∪ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})) = {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} → (⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ ∈ ({⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ∪ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})) ↔ ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ ∈ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}))
108	106, 107	mpbii 224	. . . . . . . . . . . 12 ⊢ (({⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ∪ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})) = {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} → ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ ∈ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩})
109	103	elpr 4404	. . . . . . . . . . . . 13 ⊢ (⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ ∈ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ↔ (⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩ ∨ ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩))
110		oran 1003	. . . . . . . . . . . . 13 ⊢ ((⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩ ∨ ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩) ↔ ¬ (¬ ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩ ∧ ¬ ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩))
111	109, 110	bitri 266	. . . . . . . . . . . 12 ⊢ (⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ ∈ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ↔ ¬ (¬ ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩ ∧ ¬ ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩))
112	108, 111	sylib 209	. . . . . . . . . . 11 ⊢ (({⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ∪ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})) = {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} → ¬ (¬ ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩ ∧ ¬ ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩))
113	112	necon2ai 3018	. . . . . . . . . 10 ⊢ ((¬ ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩ ∧ ¬ ⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩ = ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩) → ({⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ∪ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})) ≠ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩})
114	99, 102, 113	syl2anc 575	. . . . . . . . 9 ⊢ (𝜑 → ({⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ∪ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})) ≠ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩})
115	85, 114	eqnetrd 3056	. . . . . . . 8 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) ≠ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩})
116		fnressn 6658	. . . . . . . . . . . 12 ⊢ (((2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁) ∧ (2^nd ‘𝑇) ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇)}) = {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩})
117	67, 15, 116	syl2anc 575	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇)}) = {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩})
118		fnressn 6658	. . . . . . . . . . . 12 ⊢ (((2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁) ∧ ((2^nd ‘𝑇) + 1) ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1)}) = {⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩})
119	67, 19, 118	syl2anc 575	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1)}) = {⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩})
120	117, 119	uneq12d 3978	. . . . . . . . . 10 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇)}) ∪ ((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1)})) = ({⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩} ∪ {⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}))
121		df-pr 4384	. . . . . . . . . . . 12 ⊢ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} = ({(2^nd ‘𝑇)} ∪ {((2^nd ‘𝑇) + 1)})
122	121	reseq2i 5607	. . . . . . . . . . 11 ⊢ ((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ((2^nd ‘(1^st ‘𝑇)) ↾ ({(2^nd ‘𝑇)} ∪ {((2^nd ‘𝑇) + 1)}))
123		resundi 5627	. . . . . . . . . . 11 ⊢ ((2^nd ‘(1^st ‘𝑇)) ↾ ({(2^nd ‘𝑇)} ∪ {((2^nd ‘𝑇) + 1)})) = (((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇)}) ∪ ((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1)}))
124	122, 123	eqtri 2839	. . . . . . . . . 10 ⊢ ((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = (((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇)}) ∪ ((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1)}))
125		df-pr 4384	. . . . . . . . . 10 ⊢ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} = ({⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩} ∪ {⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩})
126	120, 124, 125	3eqtr4g 2876	. . . . . . . . 9 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩})
127		poimirlem9.4	. . . . . . . . . . 11 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)) ≠ (2^nd ‘(1^st ‘𝑇)))
128	2, 26, 49, 14, 24	poimirlem8 33748	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
129		uneq12 3972	. . . . . . . . . . . . . 14 ⊢ ((((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∧ ((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) → (((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ ((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ ((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))
130		resundi 5627	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘(1^st ‘𝑇)) ↾ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ ((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
131	22	reseq2d 5610	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ↾ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = ((2^nd ‘(1^st ‘𝑇)) ↾ (1...𝑁)))
132		fnresdm 6220	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁) → ((2^nd ‘(1^st ‘𝑇)) ↾ (1...𝑁)) = (2^nd ‘(1^st ‘𝑇)))
133	58, 66, 132	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ↾ (1...𝑁)) = (2^nd ‘(1^st ‘𝑇)))
134	131, 133	eqtrd 2851	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ↾ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (2^nd ‘(1^st ‘𝑇)))
135	130, 134	syl5eqr 2865	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ ((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (2^nd ‘(1^st ‘𝑇)))
136	39, 135	eqeq12d 2832	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ ((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ ((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) ↔ (2^nd ‘(1^st ‘𝑈)) = (2^nd ‘(1^st ‘𝑇))))
137	129, 136	syl5ib 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∧ ((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) → (2^nd ‘(1^st ‘𝑈)) = (2^nd ‘(1^st ‘𝑇))))
138	128, 137	mpan2d 677	. . . . . . . . . . . 12 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) → (2^nd ‘(1^st ‘𝑈)) = (2^nd ‘(1^st ‘𝑇))))
139	138	necon3d 3010	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) ≠ (2^nd ‘(1^st ‘𝑇)) → ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ≠ ((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
140	127, 139	mpd 15	. . . . . . . . . 10 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ≠ ((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
141	140	necomd 3044	. . . . . . . . 9 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ≠ ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
142	126, 141	eqnetrrd 3057	. . . . . . . 8 ⊢ (𝜑 → {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ≠ ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
143		prex 5112	. . . . . . . . . . . 12 ⊢ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∈ V
144	55, 143	coex 7357	. . . . . . . . . . 11 ⊢ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) ∈ V
145		prex 5112	. . . . . . . . . . 11 ⊢ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ∈ V
146	31	resex 5661	. . . . . . . . . . 11 ⊢ ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∈ V
147		hashtpg 13503	. . . . . . . . . . 11 ⊢ ((((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) ∈ V ∧ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ∈ V ∧ ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∈ V) → ((((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) ≠ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ∧ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ≠ ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∧ ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ≠ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})) ↔ (♯‘{((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})}) = 3))
148	144, 145, 146, 147	mp3an 1578	. . . . . . . . . 10 ⊢ ((((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) ≠ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ∧ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ≠ ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∧ ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ≠ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})) ↔ (♯‘{((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})}) = 3)
149	148	biimpi 207	. . . . . . . . 9 ⊢ ((((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) ≠ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ∧ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ≠ ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∧ ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ≠ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})) → (♯‘{((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})}) = 3)
150	149	3expia 1143	. . . . . . . 8 ⊢ ((((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) ≠ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ∧ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} ≠ ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) → (((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ≠ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) → (♯‘{((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})}) = 3))
151	115, 142, 150	syl2anc 575	. . . . . . 7 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ≠ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) → (♯‘{((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})}) = 3))
152		prex 5112	. . . . . . . . . . . . 13 ⊢ {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ∈ V
153		prex 5112	. . . . . . . . . . . . 13 ⊢ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∈ V
154	152, 153	mapval 8113	. . . . . . . . . . . 12 ⊢ ({((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ↑_𝑚 {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}}
155		prfi 8483	. . . . . . . . . . . . 13 ⊢ {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ∈ Fin
156		prfi 8483	. . . . . . . . . . . . 13 ⊢ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∈ Fin
157		mapfi 8510	. . . . . . . . . . . . 13 ⊢ (({((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ∈ Fin ∧ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∈ Fin) → ({((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ↑_𝑚 {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∈ Fin)
158	155, 156, 157	mp2an 675	. . . . . . . . . . . 12 ⊢ ({((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ↑_𝑚 {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∈ Fin
159	154, 158	eqeltrri 2893	. . . . . . . . . . 11 ⊢ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}} ∈ Fin
160		f1of 6362	. . . . . . . . . . . 12 ⊢ (𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} → 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))})
161	160	ss2abi 3882	. . . . . . . . . . 11 ⊢ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}} ⊆ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}}
162		ssfi 8428	. . . . . . . . . . 11 ⊢ (({𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}} ∈ Fin ∧ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}} ⊆ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}}) → {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}} ∈ Fin)
163	159, 161, 162	mp2an 675	. . . . . . . . . 10 ⊢ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}} ∈ Fin
164	19, 15	prssd 4554	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} ⊆ (1...𝑁))
165		f1ores 6376	. . . . . . . . . . . . . . 15 ⊢ (((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1→(1...𝑁) ∧ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} ⊆ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}):{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}–1-1-onto→((2^nd ‘(1^st ‘𝑇)) “ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}))
166	60, 164, 165	syl2anc 575	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}):{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}–1-1-onto→((2^nd ‘(1^st ‘𝑇)) “ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}))
167		fnimapr 6492	. . . . . . . . . . . . . . . 16 ⊢ (((2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁) ∧ ((2^nd ‘𝑇) + 1) ∈ (1...𝑁) ∧ (2^nd ‘𝑇) ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑇)) “ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}) = {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))})
168	67, 19, 15, 167	syl3anc 1483	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}) = {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))})
169	168	f1oeq3d 6359	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}):{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}–1-1-onto→((2^nd ‘(1^st ‘𝑇)) “ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}) ↔ ((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}):{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}))
170	166, 169	mpbid 223	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}):{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))})
171		f1oprg 6406	. . . . . . . . . . . . . . 15 ⊢ ((((2^nd ‘𝑇) ∈ V ∧ ((2^nd ‘𝑇) + 1) ∈ V) ∧ (((2^nd ‘𝑇) + 1) ∈ V ∧ (2^nd ‘𝑇) ∈ V)) → (((2^nd ‘𝑇) ≠ ((2^nd ‘𝑇) + 1) ∧ ((2^nd ‘𝑇) + 1) ≠ (2^nd ‘𝑇)) → {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}))
172	73, 74, 74, 73, 171	mp4an 676	. . . . . . . . . . . . . 14 ⊢ (((2^nd ‘𝑇) ≠ ((2^nd ‘𝑇) + 1) ∧ ((2^nd ‘𝑇) + 1) ≠ (2^nd ‘𝑇)) → {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)})
173	90, 91, 172	syl2anc 575	. . . . . . . . . . . . 13 ⊢ (𝜑 → {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)})
174		f1oco 6384	. . . . . . . . . . . . 13 ⊢ ((((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}):{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ∧ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}) → (((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))})
175	170, 173, 174	syl2anc 575	. . . . . . . . . . . 12 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))})
176		rnpropg 5840	. . . . . . . . . . . . . . 15 ⊢ (((2^nd ‘𝑇) ∈ V ∧ ((2^nd ‘𝑇) + 1) ∈ V) → ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} = {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)})
177	73, 74, 176	mp2an 675	. . . . . . . . . . . . . 14 ⊢ ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} = {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}
178	177	eqimssi 3867	. . . . . . . . . . . . 13 ⊢ ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ⊆ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}
179		cores 5865	. . . . . . . . . . . . 13 ⊢ (ran {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ⊆ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)} → (((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) = ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}))
180		f1oeq1 6352	. . . . . . . . . . . . 13 ⊢ ((((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) = ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) → ((((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ↔ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}))
181	178, 179, 180	mp2b 10	. . . . . . . . . . . 12 ⊢ ((((2^nd ‘(1^st ‘𝑇)) ↾ {((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)}) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ↔ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))})
182	175, 181	sylib 209	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))})
183	95	necomd 3044	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) ≠ ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)))
184		fvex 6430	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) ∈ V
185		f1oprg 6406	. . . . . . . . . . . . . 14 ⊢ ((((2^nd ‘𝑇) ∈ V ∧ ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) ∈ V) ∧ (((2^nd ‘𝑇) + 1) ∈ V ∧ ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)) ∈ V)) → (((2^nd ‘𝑇) ≠ ((2^nd ‘𝑇) + 1) ∧ ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) ≠ ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))) → {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))}))
186	73, 184, 74, 79, 185	mp4an 676	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘𝑇) ≠ ((2^nd ‘𝑇) + 1) ∧ ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) ≠ ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))) → {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))})
187	90, 183, 186	syl2anc 575	. . . . . . . . . . . 12 ⊢ (𝜑 → {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))})
188		prcom 4469	. . . . . . . . . . . . 13 ⊢ {((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))} = {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}
189		f1oeq3 6354	. . . . . . . . . . . . 13 ⊢ ({((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))} = {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} → ({⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))} ↔ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}))
190	188, 189	ax-mp 5	. . . . . . . . . . . 12 ⊢ ({⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))} ↔ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))})
191	187, 190	sylib 209	. . . . . . . . . . 11 ⊢ (𝜑 → {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))})
192		f1of1 6361	. . . . . . . . . . . . . 14 ⊢ ((2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1→(1...𝑁))
193	34, 192	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1→(1...𝑁))
194		f1ores 6376	. . . . . . . . . . . . 13 ⊢ (((2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1→(1...𝑁) ∧ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ⊆ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→((2^nd ‘(1^st ‘𝑈)) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
195	193, 20, 194	syl2anc 575	. . . . . . . . . . . 12 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→((2^nd ‘(1^st ‘𝑈)) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
196		dff1o3 6368	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2^nd ‘(1^st ‘𝑈)):(1...𝑁)–onto→(1...𝑁) ∧ Fun ^◡(2^nd ‘(1^st ‘𝑈))))
197	196	simprbi 486	. . . . . . . . . . . . . . . 16 ⊢ ((2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡(2^nd ‘(1^st ‘𝑈)))
198		imadif 6193	. . . . . . . . . . . . . . . 16 ⊢ (Fun ^◡(2^nd ‘(1^st ‘𝑈)) → ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑁) ∖ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))
199	34, 197, 198	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑁) ∖ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))
200		f1ofo 6369	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑈)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑈)):(1...𝑁)–onto→(1...𝑁))
201		foima 6345	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑈)):(1...𝑁)–onto→(1...𝑁) → ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑁)) = (1...𝑁))
202	34, 200, 201	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑁)) = (1...𝑁))
203		f1ofo 6369	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁))
204		foima 6345	. . . . . . . . . . . . . . . . . 18 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
205	58, 203, 204	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) = (1...𝑁))
206	202, 205	eqtr4d 2854	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) “ (1...𝑁)) = ((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)))
207	128	rneqd 5567	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ran ((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ran ((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
208		df-ima 5337	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ran ((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
209		df-ima 5337	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ran ((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
210	207, 208, 209	3eqtr4g 2876	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
211	206, 210	difeq12d 3939	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑈)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))
212		dff1o3 6368	. . . . . . . . . . . . . . . . . 18 ⊢ ((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))))
213	212	simprbi 486	. . . . . . . . . . . . . . . . 17 ⊢ ((2^nd ‘(1^st ‘𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ^◡(2^nd ‘(1^st ‘𝑇)))
214		imadif 6193	. . . . . . . . . . . . . . . . 17 ⊢ (Fun ^◡(2^nd ‘(1^st ‘𝑇)) → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))
215	58, 213, 214	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))
216		dfin4 4080	. . . . . . . . . . . . . . . . . 18 ⊢ ((1...𝑁) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ((1...𝑁) ∖ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
217		sseqin2 4027	. . . . . . . . . . . . . . . . . . 19 ⊢ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ⊆ (1...𝑁) ↔ ((1...𝑁) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
218	20, 217	sylib 209	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((1...𝑁) ∩ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
219	216, 218	syl5eqr 2865	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((1...𝑁) ∖ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
220	219	imaeq2d 5689	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = ((2^nd ‘(1^st ‘𝑇)) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
221	215, 220	eqtr3d 2853	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑇)) “ (1...𝑁)) ∖ ((2^nd ‘(1^st ‘𝑇)) “ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = ((2^nd ‘(1^st ‘𝑇)) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
222	199, 211, 221	3eqtrd 2855	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑁) ∖ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = ((2^nd ‘(1^st ‘𝑇)) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
223	219	imaeq2d 5689	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) “ ((1...𝑁) ∖ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = ((2^nd ‘(1^st ‘𝑈)) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
224		fnimapr 6492	. . . . . . . . . . . . . . . 16 ⊢ (((2^nd ‘(1^st ‘𝑇)) Fn (1...𝑁) ∧ (2^nd ‘𝑇) ∈ (1...𝑁) ∧ ((2^nd ‘𝑇) + 1) ∈ (1...𝑁)) → ((2^nd ‘(1^st ‘𝑇)) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = {((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))})
225	67, 15, 19, 224	syl3anc 1483	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = {((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))})
226	225, 188	syl6eq 2867	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑇)) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))})
227	222, 223, 226	3eqtr3d 2859	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))})
228	227	f1oeq3d 6359	. . . . . . . . . . . 12 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→((2^nd ‘(1^st ‘𝑈)) “ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ↔ ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}))
229	195, 228	mpbid 223	. . . . . . . . . . 11 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))})
230		ssabral 3881	. . . . . . . . . . . 12 ⊢ ({((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})} ⊆ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}} ↔ ∀𝑓 ∈ {((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})}𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))})
231		f1oeq1 6352	. . . . . . . . . . . . 13 ⊢ (𝑓 = ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) → (𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ↔ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}))
232		f1oeq1 6352	. . . . . . . . . . . . 13 ⊢ (𝑓 = {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩} → (𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ↔ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}))
233		f1oeq1 6352	. . . . . . . . . . . . 13 ⊢ (𝑓 = ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) → (𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ↔ ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}))
234	144, 145, 146, 231, 232, 233	raltp 4443	. . . . . . . . . . . 12 ⊢ (∀𝑓 ∈ {((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})}𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ↔ (((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ∧ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ∧ ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}))
235	230, 234	bitri 266	. . . . . . . . . . 11 ⊢ ({((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})} ⊆ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}} ↔ (((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ∧ {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ∧ ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}):{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}))
236	182, 191, 229, 235	syl3anbrc 1436	. . . . . . . . . 10 ⊢ (𝜑 → {((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})} ⊆ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}})
237		hashss 13433	. . . . . . . . . 10 ⊢ (({𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}} ∈ Fin ∧ {((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})} ⊆ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}}) → (♯‘{((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})}) ≤ (♯‘{𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}}))
238	163, 236, 237	sylancr 577	. . . . . . . . 9 ⊢ (𝜑 → (♯‘{((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})}) ≤ (♯‘{𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}}))
239	153	enref 8234	. . . . . . . . . . . 12 ⊢ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ≈ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}
240		hashprg 13419	. . . . . . . . . . . . . . . 16 ⊢ ((((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)) ∈ V ∧ ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) ∈ V) → (((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)) ≠ ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) ↔ (♯‘{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}) = 2))
241	79, 184, 240	mp2an 675	. . . . . . . . . . . . . . 15 ⊢ (((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)) ≠ ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇)) ↔ (♯‘{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}) = 2)
242	95, 241	sylib 209	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}) = 2)
243		hashprg 13419	. . . . . . . . . . . . . . . 16 ⊢ (((2^nd ‘𝑇) ∈ V ∧ ((2^nd ‘𝑇) + 1) ∈ V) → ((2^nd ‘𝑇) ≠ ((2^nd ‘𝑇) + 1) ↔ (♯‘{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = 2))
244	73, 74, 243	mp2an 675	. . . . . . . . . . . . . . 15 ⊢ ((2^nd ‘𝑇) ≠ ((2^nd ‘𝑇) + 1) ↔ (♯‘{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = 2)
245	90, 244	sylib 209	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = 2)
246	242, 245	eqtr4d 2854	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}) = (♯‘{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
247		hashen 13374	. . . . . . . . . . . . . 14 ⊢ (({((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ∈ Fin ∧ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∈ Fin) → ((♯‘{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}) = (♯‘{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ↔ {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ≈ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
248	155, 156, 247	mp2an 675	. . . . . . . . . . . . 13 ⊢ ((♯‘{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}) = (♯‘{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ↔ {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ≈ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
249	246, 248	sylib 209	. . . . . . . . . . . 12 ⊢ (𝜑 → {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ≈ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
250		hashfacen 13474	. . . . . . . . . . . 12 ⊢ (({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ≈ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∧ {((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))} ≈ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) → {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}} ≈ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}})
251	239, 249, 250	sylancr 577	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}} ≈ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}})
252	153, 153	mapval 8113	. . . . . . . . . . . . . 14 ⊢ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ↑_𝑚 {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}}
253		mapfi 8510	. . . . . . . . . . . . . . 15 ⊢ (({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∈ Fin ∧ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∈ Fin) → ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ↑_𝑚 {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∈ Fin)
254	156, 156, 253	mp2an 675	. . . . . . . . . . . . . 14 ⊢ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ↑_𝑚 {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∈ Fin
255	252, 254	eqeltrri 2893	. . . . . . . . . . . . 13 ⊢ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}} ∈ Fin
256		f1of 6362	. . . . . . . . . . . . . 14 ⊢ (𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} → 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})
257	256	ss2abi 3882	. . . . . . . . . . . . 13 ⊢ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}} ⊆ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}}
258		ssfi 8428	. . . . . . . . . . . . 13 ⊢ (({𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}} ∈ Fin ∧ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}} ⊆ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}⟶{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}}) → {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}} ∈ Fin)
259	255, 257, 258	mp2an 675	. . . . . . . . . . . 12 ⊢ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}} ∈ Fin
260		hashen 13374	. . . . . . . . . . . 12 ⊢ (({𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}} ∈ Fin ∧ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}} ∈ Fin) → ((♯‘{𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}}) = (♯‘{𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}}) ↔ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}} ≈ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}}))
261	163, 259, 260	mp2an 675	. . . . . . . . . . 11 ⊢ ((♯‘{𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}}) = (♯‘{𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}}) ↔ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}} ≈ {𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}})
262	251, 261	sylibr 225	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘{𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}}) = (♯‘{𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}}))
263		hashfac 13478	. . . . . . . . . . . 12 ⊢ ({(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)} ∈ Fin → (♯‘{𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}}) = (!‘(♯‘{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))
264	156, 263	ax-mp 5	. . . . . . . . . . 11 ⊢ (♯‘{𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}}) = (!‘(♯‘{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
265	245	fveq2d 6421	. . . . . . . . . . . 12 ⊢ (𝜑 → (!‘(♯‘{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = (!‘2))
266		fac2 13305	. . . . . . . . . . . 12 ⊢ (!‘2) = 2
267	265, 266	syl6eq 2867	. . . . . . . . . . 11 ⊢ (𝜑 → (!‘(♯‘{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = 2)
268	264, 267	syl5eq 2863	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘{𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}}) = 2)
269	262, 268	eqtrd 2851	. . . . . . . . 9 ⊢ (𝜑 → (♯‘{𝑓 ∣ 𝑓:{(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}–1-1-onto→{((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1)), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))}}) = 2)
270	238, 269	breqtrd 4881	. . . . . . . 8 ⊢ (𝜑 → (♯‘{((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})}) ≤ 2)
271		breq1 4858	. . . . . . . 8 ⊢ ((♯‘{((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})}) = 3 → ((♯‘{((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})}) ≤ 2 ↔ 3 ≤ 2))
272	270, 271	syl5ibcom 236	. . . . . . 7 ⊢ (𝜑 → ((♯‘{((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}), {⟨(2^nd ‘𝑇), ((2^nd ‘(1^st ‘𝑇))‘(2^nd ‘𝑇))⟩, ⟨((2^nd ‘𝑇) + 1), ((2^nd ‘(1^st ‘𝑇))‘((2^nd ‘𝑇) + 1))⟩}, ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})}) = 3 → 3 ≤ 2))
273	151, 272	syld 47	. . . . . 6 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ≠ ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) → 3 ≤ 2))
274	273	necon1bd 3007	. . . . 5 ⊢ (𝜑 → (¬ 3 ≤ 2 → ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩})))
275	44, 274	mpi 20	. . . 4 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) = ((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}))
276		coires1 5880	. . . . 5 ⊢ ((2^nd ‘(1^st ‘𝑇)) ∘ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = ((2^nd ‘(1^st ‘𝑇)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))
277	128, 276	syl6eqr 2869	. . . 4 ⊢ (𝜑 → ((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})) = ((2^nd ‘(1^st ‘𝑇)) ∘ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))
278	275, 277	uneq12d 3978	. . 3 ⊢ (𝜑 → (((2^nd ‘(1^st ‘𝑈)) ↾ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}) ∪ ((2^nd ‘(1^st ‘𝑈)) ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))) = (((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) ∪ ((2^nd ‘(1^st ‘𝑇)) ∘ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))))
279	39, 278	eqtr3d 2853	. 2 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)) = (((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) ∪ ((2^nd ‘(1^st ‘𝑇)) ∘ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))))
280		coundi 5863	. 2 ⊢ ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))) = (((2^nd ‘(1^st ‘𝑇)) ∘ {⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩}) ∪ ((2^nd ‘(1^st ‘𝑇)) ∘ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)}))))
281	279, 280	syl6eqr 2869	1 ⊢ (𝜑 → (2^nd ‘(1^st ‘𝑈)) = ((2^nd ‘(1^st ‘𝑇)) ∘ ({⟨(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)⟩, ⟨((2^nd ‘𝑇) + 1), (2^nd ‘𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2^nd ‘𝑇), ((2^nd ‘𝑇) + 1)})))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 197 ∧ wa 384 ∨ wo 865 ∧ w3a 1100 = wceq 1637 ∈ wcel 2157 {cab 2803 ≠ wne 2989 ∀wral 3107 {crab 3111 Vcvv 3402 ⦋csb 3739 ∖ cdif 3777 ∪ cun 3778 ∩ cin 3779 ⊆ wss 3780 ifcif 4290 {csn 4381 {cpr 4383 {ctp 4385 ⟨cop 4387 class class class wbr 4855 ↦ cmpt 4934 I cid 5231 × cxp 5322 ^◡ccnv 5323 ran crn 5325 ↾ cres 5326 “ cima 5327 ∘ ccom 5328 Fun wfun 6104 Fn wfn 6105 ⟶wf 6106 –1-1→wf1 6107 –onto→wfo 6108 –1-1-onto→wf1o 6109 ‘cfv 6110 (class class class)co 6883 ∘_𝑓 cof 7134 1^st c1st 7405 2^nd c2nd 7406 ↑_𝑚 cmap 8101 ≈ cen 8198 Fincfn 8201 ℂcc 10228 0cc0 10230 1c1 10231 + caddc 10233 < clt 10368 ≤ cle 10369 − cmin 10560 ℕcn 11314 2c2 11367 3c3 11368 ℤcz 11662 ℤ_≥cuz 11923 ...cfz 12568 ..^cfzo 12708 !cfa 13299 ♯chash 13356

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1877 ax-4 1894 ax-5 2001 ax-6 2069 ax-7 2105 ax-8 2159 ax-9 2166 ax-10 2186 ax-11 2202 ax-12 2215 ax-13 2422 ax-ext 2795 ax-rep 4977 ax-sep 4988 ax-nul 4996 ax-pow 5048 ax-pr 5109 ax-un 7188 ax-cnex 10286 ax-resscn 10287 ax-1cn 10288 ax-icn 10289 ax-addcl 10290 ax-addrcl 10291 ax-mulcl 10292 ax-mulrcl 10293 ax-mulcom 10294 ax-addass 10295 ax-mulass 10296 ax-distr 10297 ax-i2m1 10298 ax-1ne0 10299 ax-1rid 10300 ax-rnegex 10301 ax-rrecex 10302 ax-cnre 10303 ax-pre-lttri 10304 ax-pre-lttrn 10305 ax-pre-ltadd 10306 ax-pre-mulgt0 10307

This theorem depends on definitions: df-bi 198 df-an 385 df-or 866 df-3or 1101 df-3an 1102 df-tru 1641 df-ex 1860 df-nf 1864 df-sb 2062 df-mo 2635 df-eu 2642 df-clab 2804 df-cleq 2810 df-clel 2813 df-nfc 2948 df-ne 2990 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3404 df-sbc 3645 df-csb 3740 df-dif 3783 df-un 3785 df-in 3787 df-ss 3794 df-pss 3796 df-nul 4128 df-if 4291 df-pw 4364 df-sn 4382 df-pr 4384 df-tp 4386 df-op 4388 df-uni 4642 df-int 4681 df-iun 4725 df-br 4856 df-opab 4918 df-mpt 4935 df-tr 4958 df-id 5232 df-eprel 5237 df-po 5245 df-so 5246 df-fr 5283 df-we 5285 df-xp 5330 df-rel 5331 df-cnv 5332 df-co 5333 df-dm 5334 df-rn 5335 df-res 5336 df-ima 5337 df-pred 5906 df-ord 5952 df-on 5953 df-lim 5954 df-suc 5955 df-iota 6073 df-fun 6112 df-fn 6113 df-f 6114 df-f1 6115 df-fo 6116 df-f1o 6117 df-fv 6118 df-riota 6844 df-ov 6886 df-oprab 6887 df-mpt2 6888 df-of 7136 df-om 7305 df-1st 7407 df-2nd 7408 df-wrecs 7651 df-recs 7713 df-rdg 7751 df-1o 7805 df-2o 7806 df-oadd 7809 df-er 7988 df-map 8103 df-pm 8104 df-en 8202 df-dom 8203 df-sdom 8204 df-fin 8205 df-card 9057 df-cda 9284 df-pnf 10370 df-mnf 10371 df-xr 10372 df-ltxr 10373 df-le 10374 df-sub 10562 df-neg 10563 df-div 10979 df-nn 11315 df-2 11375 df-3 11376 df-n0 11579 df-xnn0 11649 df-z 11663 df-uz 11924 df-fz 12569 df-fzo 12709 df-seq 13044 df-fac 13300 df-bc 13329 df-hash 13357

This theorem is referenced by: poimirlem22 33762

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