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Theorem infxpenc 9127

Description: A canonical version of infxpen 9123, by a completely different approach (although it uses infxpen 9123 via xpomen 9124). Using Cantor's normal form, we can show that 𝐴 ↑_𝑜 𝐵 respects equinumerosity (oef1o 8845), so that all the steps of (ω↑𝑊) · (ω↑𝑊) ≈ ω↑(2𝑊) ≈ (ω↑2)↑𝑊 ≈ ω↑𝑊 can be verified using bijections to do the ordinal commutations. (The assumption on 𝑁 can be satisfied using cnfcom3c 8853.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 7-Jul-2019.)

**Hypotheses**
Ref	Expression
infxpenc.1	⊢ (𝜑 → 𝐴 ∈ On)
infxpenc.2	⊢ (𝜑 → ω ⊆ 𝐴)
infxpenc.3	⊢ (𝜑 → 𝑊 ∈ (On ∖ 1_𝑜))
infxpenc.4	⊢ (𝜑 → 𝐹:(ω ↑_𝑜 2_𝑜)–1-1-onto→ω)
infxpenc.5	⊢ (𝜑 → (𝐹‘∅) = ∅)
infxpenc.6	⊢ (𝜑 → 𝑁:𝐴–1-1-onto→(ω ↑_𝑜 𝑊))
infxpenc.k	⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑_𝑜 2_𝑜) ↑_𝑚 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ^◡( I ↾ 𝑊))))
infxpenc.h	⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ^◡((ω ↑_𝑜 2_𝑜) CNF 𝑊))
infxpenc.l	⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑_𝑚 (𝑊 ·_𝑜 2_𝑜)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ^◡(𝑌 ∘ ^◡𝑋))))
infxpenc.x	⊢ 𝑋 = (𝑧 ∈ 2_𝑜, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·_𝑜 𝑧) +_𝑜 𝑤))
infxpenc.y	⊢ 𝑌 = (𝑧 ∈ 2_𝑜, 𝑤 ∈ 𝑊 ↦ ((2_𝑜 ·_𝑜 𝑤) +_𝑜 𝑧))
infxpenc.j	⊢ 𝐽 = (((ω CNF (2_𝑜 ·_𝑜 𝑊)) ∘ 𝐿) ∘ ^◡(ω CNF (𝑊 ·_𝑜 2_𝑜)))
infxpenc.z	⊢ 𝑍 = (𝑥 ∈ (ω ↑_𝑜 𝑊), 𝑦 ∈ (ω ↑_𝑜 𝑊) ↦ (((ω ↑_𝑜 𝑊) ·_𝑜 𝑥) +_𝑜 𝑦))
infxpenc.t	⊢ 𝑇 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩)
infxpenc.g	⊢ 𝐺 = (^◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇))

**Assertion**
Ref	Expression
infxpenc	⊢ (𝜑 → 𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴)

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐹,𝑦 𝑥,𝑁,𝑦 𝜑,𝑥,𝑦 𝑥,𝑤,𝑦,𝑧,𝑊 𝑥,𝑋,𝑦 𝑥,𝑌,𝑦 Allowed substitution hints: 𝜑(𝑧,𝑤) 𝐴(𝑧,𝑤) 𝑇(𝑥,𝑦,𝑧,𝑤) 𝐹(𝑧,𝑤) 𝐺(𝑥,𝑦,𝑧,𝑤) 𝐻(𝑥,𝑦,𝑧,𝑤) 𝐽(𝑥,𝑦,𝑧,𝑤) 𝐾(𝑥,𝑦,𝑧,𝑤) 𝐿(𝑥,𝑦,𝑧,𝑤) 𝑁(𝑧,𝑤) 𝑋(𝑧,𝑤) 𝑌(𝑧,𝑤) 𝑍(𝑥,𝑦,𝑧,𝑤)

**Proof of Theorem infxpenc**
Step	Hyp	Ref	Expression
1		infxpenc.6	. . . 4 ⊢ (𝜑 → 𝑁:𝐴–1-1-onto→(ω ↑_𝑜 𝑊))
2		f1ocnv 6368	. . . 4 ⊢ (𝑁:𝐴–1-1-onto→(ω ↑_𝑜 𝑊) → ^◡𝑁:(ω ↑_𝑜 𝑊)–1-1-onto→𝐴)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → ^◡𝑁:(ω ↑_𝑜 𝑊)–1-1-onto→𝐴)
4		infxpenc.4	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(ω ↑_𝑜 2_𝑜)–1-1-onto→ω)
5		f1oi 6393	. . . . . . . . 9 ⊢ ( I ↾ 𝑊):𝑊–1-1-onto→𝑊
6	5	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ( I ↾ 𝑊):𝑊–1-1-onto→𝑊)
7		omelon 8793	. . . . . . . . . . 11 ⊢ ω ∈ On
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ω ∈ On)
9		2on 7808	. . . . . . . . . 10 ⊢ 2_𝑜 ∈ On
10		oecl 7857	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 2_𝑜 ∈ On) → (ω ↑_𝑜 2_𝑜) ∈ On)
11	8, 9, 10	sylancl 581	. . . . . . . . 9 ⊢ (𝜑 → (ω ↑_𝑜 2_𝑜) ∈ On)
12	9	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2_𝑜 ∈ On)
13		peano1 7319	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
14	13	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ ω)
15		oen0 7906	. . . . . . . . . 10 ⊢ (((ω ∈ On ∧ 2_𝑜 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑_𝑜 2_𝑜))
16	8, 12, 14, 15	syl21anc 867	. . . . . . . . 9 ⊢ (𝜑 → ∅ ∈ (ω ↑_𝑜 2_𝑜))
17		ondif1 7821	. . . . . . . . 9 ⊢ ((ω ↑_𝑜 2_𝑜) ∈ (On ∖ 1_𝑜) ↔ ((ω ↑_𝑜 2_𝑜) ∈ On ∧ ∅ ∈ (ω ↑_𝑜 2_𝑜)))
18	11, 16, 17	sylanbrc 579	. . . . . . . 8 ⊢ (𝜑 → (ω ↑_𝑜 2_𝑜) ∈ (On ∖ 1_𝑜))
19		infxpenc.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1_𝑜))
20	19	eldifad 3781	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ On)
21		infxpenc.5	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘∅) = ∅)
22		infxpenc.k	. . . . . . . 8 ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑_𝑜 2_𝑜) ↑_𝑚 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ^◡( I ↾ 𝑊))))
23		infxpenc.h	. . . . . . . 8 ⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ^◡((ω ↑_𝑜 2_𝑜) CNF 𝑊))
24	4, 6, 18, 20, 8, 20, 21, 22, 23	oef1o 8845	. . . . . . 7 ⊢ (𝜑 → 𝐻:((ω ↑_𝑜 2_𝑜) ↑_𝑜 𝑊)–1-1-onto→(ω ↑_𝑜 𝑊))
25		f1oi 6393	. . . . . . . . . 10 ⊢ ( I ↾ ω):ω–1-1-onto→ω
26	25	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ( I ↾ ω):ω–1-1-onto→ω)
27		infxpenc.x	. . . . . . . . . . 11 ⊢ 𝑋 = (𝑧 ∈ 2_𝑜, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·_𝑜 𝑧) +_𝑜 𝑤))
28		infxpenc.y	. . . . . . . . . . 11 ⊢ 𝑌 = (𝑧 ∈ 2_𝑜, 𝑤 ∈ 𝑊 ↦ ((2_𝑜 ·_𝑜 𝑤) +_𝑜 𝑧))
29	27, 28	omf1o 8305	. . . . . . . . . 10 ⊢ ((𝑊 ∈ On ∧ 2_𝑜 ∈ On) → (𝑌 ∘ ^◡𝑋):(𝑊 ·_𝑜 2_𝑜)–1-1-onto→(2_𝑜 ·_𝑜 𝑊))
30	20, 9, 29	sylancl 581	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∘ ^◡𝑋):(𝑊 ·_𝑜 2_𝑜)–1-1-onto→(2_𝑜 ·_𝑜 𝑊))
31		ondif1 7821	. . . . . . . . . . 11 ⊢ (ω ∈ (On ∖ 1_𝑜) ↔ (ω ∈ On ∧ ∅ ∈ ω))
32	7, 13, 31	mpbir2an 703	. . . . . . . . . 10 ⊢ ω ∈ (On ∖ 1_𝑜)
33	32	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ω ∈ (On ∖ 1_𝑜))
34		omcl 7856	. . . . . . . . . 10 ⊢ ((𝑊 ∈ On ∧ 2_𝑜 ∈ On) → (𝑊 ·_𝑜 2_𝑜) ∈ On)
35	20, 9, 34	sylancl 581	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ·_𝑜 2_𝑜) ∈ On)
36		omcl 7856	. . . . . . . . . 10 ⊢ ((2_𝑜 ∈ On ∧ 𝑊 ∈ On) → (2_𝑜 ·_𝑜 𝑊) ∈ On)
37	12, 20, 36	syl2anc 580	. . . . . . . . 9 ⊢ (𝜑 → (2_𝑜 ·_𝑜 𝑊) ∈ On)
38		fvresi 6668	. . . . . . . . . 10 ⊢ (∅ ∈ ω → (( I ↾ ω)‘∅) = ∅)
39	13, 38	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → (( I ↾ ω)‘∅) = ∅)
40		infxpenc.l	. . . . . . . . 9 ⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑_𝑚 (𝑊 ·_𝑜 2_𝑜)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ^◡(𝑌 ∘ ^◡𝑋))))
41		infxpenc.j	. . . . . . . . 9 ⊢ 𝐽 = (((ω CNF (2_𝑜 ·_𝑜 𝑊)) ∘ 𝐿) ∘ ^◡(ω CNF (𝑊 ·_𝑜 2_𝑜)))
42	26, 30, 33, 35, 8, 37, 39, 40, 41	oef1o 8845	. . . . . . . 8 ⊢ (𝜑 → 𝐽:(ω ↑_𝑜 (𝑊 ·_𝑜 2_𝑜))–1-1-onto→(ω ↑_𝑜 (2_𝑜 ·_𝑜 𝑊)))
43		oeoe 7919	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 2_𝑜 ∈ On ∧ 𝑊 ∈ On) → ((ω ↑_𝑜 2_𝑜) ↑_𝑜 𝑊) = (ω ↑_𝑜 (2_𝑜 ·_𝑜 𝑊)))
44	8, 12, 20, 43	syl3anc 1491	. . . . . . . . 9 ⊢ (𝜑 → ((ω ↑_𝑜 2_𝑜) ↑_𝑜 𝑊) = (ω ↑_𝑜 (2_𝑜 ·_𝑜 𝑊)))
45		f1oeq3 6347	. . . . . . . . 9 ⊢ (((ω ↑_𝑜 2_𝑜) ↑_𝑜 𝑊) = (ω ↑_𝑜 (2_𝑜 ·_𝑜 𝑊)) → (𝐽:(ω ↑_𝑜 (𝑊 ·_𝑜 2_𝑜))–1-1-onto→((ω ↑_𝑜 2_𝑜) ↑_𝑜 𝑊) ↔ 𝐽:(ω ↑_𝑜 (𝑊 ·_𝑜 2_𝑜))–1-1-onto→(ω ↑_𝑜 (2_𝑜 ·_𝑜 𝑊))))
46	44, 45	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐽:(ω ↑_𝑜 (𝑊 ·_𝑜 2_𝑜))–1-1-onto→((ω ↑_𝑜 2_𝑜) ↑_𝑜 𝑊) ↔ 𝐽:(ω ↑_𝑜 (𝑊 ·_𝑜 2_𝑜))–1-1-onto→(ω ↑_𝑜 (2_𝑜 ·_𝑜 𝑊))))
47	42, 46	mpbird 249	. . . . . . 7 ⊢ (𝜑 → 𝐽:(ω ↑_𝑜 (𝑊 ·_𝑜 2_𝑜))–1-1-onto→((ω ↑_𝑜 2_𝑜) ↑_𝑜 𝑊))
48		f1oco 6378	. . . . . . 7 ⊢ ((𝐻:((ω ↑_𝑜 2_𝑜) ↑_𝑜 𝑊)–1-1-onto→(ω ↑_𝑜 𝑊) ∧ 𝐽:(ω ↑_𝑜 (𝑊 ·_𝑜 2_𝑜))–1-1-onto→((ω ↑_𝑜 2_𝑜) ↑_𝑜 𝑊)) → (𝐻 ∘ 𝐽):(ω ↑_𝑜 (𝑊 ·_𝑜 2_𝑜))–1-1-onto→(ω ↑_𝑜 𝑊))
49	24, 47, 48	syl2anc 580	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘ 𝐽):(ω ↑_𝑜 (𝑊 ·_𝑜 2_𝑜))–1-1-onto→(ω ↑_𝑜 𝑊))
50		df-2o 7800	. . . . . . . . . . . 12 ⊢ 2_𝑜 = suc 1_𝑜
51	50	oveq2i 6889	. . . . . . . . . . 11 ⊢ (𝑊 ·_𝑜 2_𝑜) = (𝑊 ·_𝑜 suc 1_𝑜)
52		1on 7806	. . . . . . . . . . . 12 ⊢ 1_𝑜 ∈ On
53		omsuc 7846	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ On ∧ 1_𝑜 ∈ On) → (𝑊 ·_𝑜 suc 1_𝑜) = ((𝑊 ·_𝑜 1_𝑜) +_𝑜 𝑊))
54	20, 52, 53	sylancl 581	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊 ·_𝑜 suc 1_𝑜) = ((𝑊 ·_𝑜 1_𝑜) +_𝑜 𝑊))
55	51, 54	syl5eq 2845	. . . . . . . . . 10 ⊢ (𝜑 → (𝑊 ·_𝑜 2_𝑜) = ((𝑊 ·_𝑜 1_𝑜) +_𝑜 𝑊))
56		om1 7862	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ On → (𝑊 ·_𝑜 1_𝑜) = 𝑊)
57	20, 56	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊 ·_𝑜 1_𝑜) = 𝑊)
58	57	oveq1d 6893	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑊 ·_𝑜 1_𝑜) +_𝑜 𝑊) = (𝑊 +_𝑜 𝑊))
59	55, 58	eqtrd 2833	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ·_𝑜 2_𝑜) = (𝑊 +_𝑜 𝑊))
60	59	oveq2d 6894	. . . . . . . 8 ⊢ (𝜑 → (ω ↑_𝑜 (𝑊 ·_𝑜 2_𝑜)) = (ω ↑_𝑜 (𝑊 +_𝑜 𝑊)))
61		oeoa 7917	. . . . . . . . 9 ⊢ ((ω ∈ On ∧ 𝑊 ∈ On ∧ 𝑊 ∈ On) → (ω ↑_𝑜 (𝑊 +_𝑜 𝑊)) = ((ω ↑_𝑜 𝑊) ·_𝑜 (ω ↑_𝑜 𝑊)))
62	8, 20, 20, 61	syl3anc 1491	. . . . . . . 8 ⊢ (𝜑 → (ω ↑_𝑜 (𝑊 +_𝑜 𝑊)) = ((ω ↑_𝑜 𝑊) ·_𝑜 (ω ↑_𝑜 𝑊)))
63	60, 62	eqtrd 2833	. . . . . . 7 ⊢ (𝜑 → (ω ↑_𝑜 (𝑊 ·_𝑜 2_𝑜)) = ((ω ↑_𝑜 𝑊) ·_𝑜 (ω ↑_𝑜 𝑊)))
64	63	f1oeq2d 6352	. . . . . 6 ⊢ (𝜑 → ((𝐻 ∘ 𝐽):(ω ↑_𝑜 (𝑊 ·_𝑜 2_𝑜))–1-1-onto→(ω ↑_𝑜 𝑊) ↔ (𝐻 ∘ 𝐽):((ω ↑_𝑜 𝑊) ·_𝑜 (ω ↑_𝑜 𝑊))–1-1-onto→(ω ↑_𝑜 𝑊)))
65	49, 64	mpbid 224	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ 𝐽):((ω ↑_𝑜 𝑊) ·_𝑜 (ω ↑_𝑜 𝑊))–1-1-onto→(ω ↑_𝑜 𝑊))
66		oecl 7857	. . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑊 ∈ On) → (ω ↑_𝑜 𝑊) ∈ On)
67	8, 20, 66	syl2anc 580	. . . . . 6 ⊢ (𝜑 → (ω ↑_𝑜 𝑊) ∈ On)
68		infxpenc.z	. . . . . . 7 ⊢ 𝑍 = (𝑥 ∈ (ω ↑_𝑜 𝑊), 𝑦 ∈ (ω ↑_𝑜 𝑊) ↦ (((ω ↑_𝑜 𝑊) ·_𝑜 𝑥) +_𝑜 𝑦))
69	68	omxpenlem 8303	. . . . . 6 ⊢ (((ω ↑_𝑜 𝑊) ∈ On ∧ (ω ↑_𝑜 𝑊) ∈ On) → 𝑍:((ω ↑_𝑜 𝑊) × (ω ↑_𝑜 𝑊))–1-1-onto→((ω ↑_𝑜 𝑊) ·_𝑜 (ω ↑_𝑜 𝑊)))
70	67, 67, 69	syl2anc 580	. . . . 5 ⊢ (𝜑 → 𝑍:((ω ↑_𝑜 𝑊) × (ω ↑_𝑜 𝑊))–1-1-onto→((ω ↑_𝑜 𝑊) ·_𝑜 (ω ↑_𝑜 𝑊)))
71		f1oco 6378	. . . . 5 ⊢ (((𝐻 ∘ 𝐽):((ω ↑_𝑜 𝑊) ·_𝑜 (ω ↑_𝑜 𝑊))–1-1-onto→(ω ↑_𝑜 𝑊) ∧ 𝑍:((ω ↑_𝑜 𝑊) × (ω ↑_𝑜 𝑊))–1-1-onto→((ω ↑_𝑜 𝑊) ·_𝑜 (ω ↑_𝑜 𝑊))) → ((𝐻 ∘ 𝐽) ∘ 𝑍):((ω ↑_𝑜 𝑊) × (ω ↑_𝑜 𝑊))–1-1-onto→(ω ↑_𝑜 𝑊))
72	65, 70, 71	syl2anc 580	. . . 4 ⊢ (𝜑 → ((𝐻 ∘ 𝐽) ∘ 𝑍):((ω ↑_𝑜 𝑊) × (ω ↑_𝑜 𝑊))–1-1-onto→(ω ↑_𝑜 𝑊))
73		f1of 6356	. . . . . . . . . 10 ⊢ (𝑁:𝐴–1-1-onto→(ω ↑_𝑜 𝑊) → 𝑁:𝐴⟶(ω ↑_𝑜 𝑊))
74	1, 73	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑁:𝐴⟶(ω ↑_𝑜 𝑊))
75	74	feqmptd 6474	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)))
76		f1oeq1 6345	. . . . . . . 8 ⊢ (𝑁 = (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)) → (𝑁:𝐴–1-1-onto→(ω ↑_𝑜 𝑊) ↔ (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)):𝐴–1-1-onto→(ω ↑_𝑜 𝑊)))
77	75, 76	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑁:𝐴–1-1-onto→(ω ↑_𝑜 𝑊) ↔ (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)):𝐴–1-1-onto→(ω ↑_𝑜 𝑊)))
78	1, 77	mpbid 224	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)):𝐴–1-1-onto→(ω ↑_𝑜 𝑊))
79	74	feqmptd 6474	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)))
80		f1oeq1 6345	. . . . . . . 8 ⊢ (𝑁 = (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)) → (𝑁:𝐴–1-1-onto→(ω ↑_𝑜 𝑊) ↔ (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)):𝐴–1-1-onto→(ω ↑_𝑜 𝑊)))
81	79, 80	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑁:𝐴–1-1-onto→(ω ↑_𝑜 𝑊) ↔ (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)):𝐴–1-1-onto→(ω ↑_𝑜 𝑊)))
82	1, 81	mpbid 224	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)):𝐴–1-1-onto→(ω ↑_𝑜 𝑊))
83	78, 82	xpf1o 8364	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩):(𝐴 × 𝐴)–1-1-onto→((ω ↑_𝑜 𝑊) × (ω ↑_𝑜 𝑊)))
84		infxpenc.t	. . . . . 6 ⊢ 𝑇 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩)
85		f1oeq1 6345	. . . . . 6 ⊢ (𝑇 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩) → (𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑_𝑜 𝑊) × (ω ↑_𝑜 𝑊)) ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩):(𝐴 × 𝐴)–1-1-onto→((ω ↑_𝑜 𝑊) × (ω ↑_𝑜 𝑊))))
86	84, 85	ax-mp 5	. . . . 5 ⊢ (𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑_𝑜 𝑊) × (ω ↑_𝑜 𝑊)) ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩):(𝐴 × 𝐴)–1-1-onto→((ω ↑_𝑜 𝑊) × (ω ↑_𝑜 𝑊)))
87	83, 86	sylibr 226	. . . 4 ⊢ (𝜑 → 𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑_𝑜 𝑊) × (ω ↑_𝑜 𝑊)))
88		f1oco 6378	. . . 4 ⊢ ((((𝐻 ∘ 𝐽) ∘ 𝑍):((ω ↑_𝑜 𝑊) × (ω ↑_𝑜 𝑊))–1-1-onto→(ω ↑_𝑜 𝑊) ∧ 𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑_𝑜 𝑊) × (ω ↑_𝑜 𝑊))) → (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑_𝑜 𝑊))
89	72, 87, 88	syl2anc 580	. . 3 ⊢ (𝜑 → (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑_𝑜 𝑊))
90		f1oco 6378	. . 3 ⊢ ((^◡𝑁:(ω ↑_𝑜 𝑊)–1-1-onto→𝐴 ∧ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑_𝑜 𝑊)) → (^◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto→𝐴)
91	3, 89, 90	syl2anc 580	. 2 ⊢ (𝜑 → (^◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto→𝐴)
92		infxpenc.g	. . 3 ⊢ 𝐺 = (^◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇))
93		f1oeq1 6345	. . 3 ⊢ (𝐺 = (^◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)) → (𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴 ↔ (^◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto→𝐴))
94	92, 93	ax-mp 5	. 2 ⊢ (𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴 ↔ (^◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto→𝐴)
95	91, 94	sylibr 226	1 ⊢ (𝜑 → 𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 = wceq 1653 ∈ wcel 2157 {crab 3093 ∖ cdif 3766 ⊆ wss 3769 ∅c0 4115 ⟨cop 4374 class class class wbr 4843 ↦ cmpt 4922 I cid 5219 × cxp 5310 ^◡ccnv 5311 ↾ cres 5314 ∘ ccom 5316 Oncon0 5941 suc csuc 5943 ⟶wf 6097 –1-1-onto→wf1o 6100 ‘cfv 6101 (class class class)co 6878 ↦ cmpt2 6880 ωcom 7299 1_𝑜c1o 7792 2_𝑜c2o 7793 +_𝑜 coa 7796 ·_𝑜 comu 7797 ↑_𝑜 coe 7798 ↑_𝑚 cmap 8095 finSupp cfsupp 8517 CNF ccnf 8808

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-inf2 8788

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-se 5272 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-isom 6110 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-supp 7533 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-seqom 7782 df-1o 7799 df-2o 7800 df-oadd 7803 df-omul 7804 df-oexp 7805 df-er 7982 df-map 8097 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-fsupp 8518 df-oi 8657 df-cnf 8809

This theorem is referenced by: infxpenc2lem2 9129

W3C validator