Theorem infxpenc 10051

Description: A canonical version of infxpen 10047, by a completely different approach (although it uses infxpen 10047 via xpomen 10048). Using Cantor's normal form, we can show that 𝐴 ↑_o 𝐵 respects equinumerosity (oef1o 9731), 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 9739.) (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 ∖ 1o))
infxpenc.4	⊢ (𝜑 → 𝐹:(ω ↑o 2o)–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 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))
infxpenc.h	⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ◡((ω ↑o 2o) CNF 𝑊))
infxpenc.l	⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡(𝑌 ∘ ◡𝑋))))
infxpenc.x	⊢ 𝑋 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))
infxpenc.y	⊢ 𝑌 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧))
infxpenc.j	⊢ 𝐽 = (((ω CNF (2o ·o 𝑊)) ∘ 𝐿) ∘ ◡(ω CNF (𝑊 ·o 2o)))
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 6856	. . . 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 2o)–1-1-onto→ω)
5		f1oi 6882	. . . . . . . . 9 ⊢ ( I ↾ 𝑊):𝑊–1-1-onto→𝑊
6	5	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ( I ↾ 𝑊):𝑊–1-1-onto→𝑊)
7		omelon 9679	. . . . . . . . . . 11 ⊢ ω ∈ On
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ω ∈ On)
9		2on 8509	. . . . . . . . . 10 ⊢ 2o ∈ On
10		oecl 8566	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 2o ∈ On) → (ω ↑o 2o) ∈ On)
11	8, 9, 10	sylancl 584	. . . . . . . . 9 ⊢ (𝜑 → (ω ↑o 2o) ∈ On)
12	9	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2o ∈ On)
13		peano1 7902	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
14	13	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ ω)
15		oen0 8615	. . . . . . . . . 10 ⊢ (((ω ∈ On ∧ 2o ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 2o))
16	8, 12, 14, 15	syl21anc 836	. . . . . . . . 9 ⊢ (𝜑 → ∅ ∈ (ω ↑o 2o))
17		ondif1 8530	. . . . . . . . 9 ⊢ ((ω ↑o 2o) ∈ (On ∖ 1o) ↔ ((ω ↑o 2o) ∈ On ∧ ∅ ∈ (ω ↑o 2o)))
18	11, 16, 17	sylanbrc 581	. . . . . . . 8 ⊢ (𝜑 → (ω ↑o 2o) ∈ (On ∖ 1o))
19		infxpenc.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1o))
20	19	eldifad 3961	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ On)
21		infxpenc.5	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘∅) = ∅)
22		infxpenc.k	. . . . . . . 8 ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ ((ω ↑o 2o) ↑m 𝑊) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡( I ↾ 𝑊))))
23		infxpenc.h	. . . . . . . 8 ⊢ 𝐻 = (((ω CNF 𝑊) ∘ 𝐾) ∘ ◡((ω ↑o 2o) CNF 𝑊))
24	4, 6, 18, 20, 8, 20, 21, 22, 23	oef1o 9731	. . . . . . 7 ⊢ (𝜑 → 𝐻:((ω ↑o 2o) ↑o 𝑊)–1-1-onto→(ω ↑o 𝑊))
25		f1oi 6882	. . . . . . . . . 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 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((𝑊 ·o 𝑧) +o 𝑤))
28		infxpenc.y	. . . . . . . . . . 11 ⊢ 𝑌 = (𝑧 ∈ 2o, 𝑤 ∈ 𝑊 ↦ ((2o ·o 𝑤) +o 𝑧))
29	27, 28	omf1o 9108	. . . . . . . . . 10 ⊢ ((𝑊 ∈ On ∧ 2o ∈ On) → (𝑌 ∘ ◡𝑋):(𝑊 ·o 2o)–1-1-onto→(2o ·o 𝑊))
30	20, 9, 29	sylancl 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∘ ◡𝑋):(𝑊 ·o 2o)–1-1-onto→(2o ·o 𝑊))
31		ondif1 8530	. . . . . . . . . . 11 ⊢ (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
32	7, 13, 31	mpbir2an 709	. . . . . . . . . 10 ⊢ ω ∈ (On ∖ 1o)
33	32	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ω ∈ (On ∖ 1o))
34		omcl 8565	. . . . . . . . . 10 ⊢ ((𝑊 ∈ On ∧ 2o ∈ On) → (𝑊 ·o 2o) ∈ On)
35	20, 9, 34	sylancl 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ·o 2o) ∈ On)
36		omcl 8565	. . . . . . . . . 10 ⊢ ((2o ∈ On ∧ 𝑊 ∈ On) → (2o ·o 𝑊) ∈ On)
37	12, 20, 36	syl2anc 582	. . . . . . . . 9 ⊢ (𝜑 → (2o ·o 𝑊) ∈ On)
38		fvresi 7188	. . . . . . . . . 10 ⊢ (∅ ∈ ω → (( I ↾ ω)‘∅) = ∅)
39	13, 38	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → (( I ↾ ω)‘∅) = ∅)
40		infxpenc.l	. . . . . . . . 9 ⊢ 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑m (𝑊 ·o 2o)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦 ∘ ◡(𝑌 ∘ ◡𝑋))))
41		infxpenc.j	. . . . . . . . 9 ⊢ 𝐽 = (((ω CNF (2o ·o 𝑊)) ∘ 𝐿) ∘ ◡(ω CNF (𝑊 ·o 2o)))
42	26, 30, 33, 35, 8, 37, 39, 40, 41	oef1o 9731	. . . . . . . 8 ⊢ (𝜑 → 𝐽:(ω ↑o (𝑊 ·o 2o))–1-1-onto→(ω ↑o (2o ·o 𝑊)))
43		oeoe 8628	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 2o ∈ On ∧ 𝑊 ∈ On) → ((ω ↑o 2o) ↑o 𝑊) = (ω ↑o (2o ·o 𝑊)))
44	7, 12, 20, 43	mp3an2i 1462	. . . . . . . . 9 ⊢ (𝜑 → ((ω ↑o 2o) ↑o 𝑊) = (ω ↑o (2o ·o 𝑊)))
45	44	f1oeq3d 6841	. . . . . . . 8 ⊢ (𝜑 → (𝐽:(ω ↑o (𝑊 ·o 2o))–1-1-onto→((ω ↑o 2o) ↑o 𝑊) ↔ 𝐽:(ω ↑o (𝑊 ·o 2o))–1-1-onto→(ω ↑o (2o ·o 𝑊))))
46	42, 45	mpbird 256	. . . . . . 7 ⊢ (𝜑 → 𝐽:(ω ↑o (𝑊 ·o 2o))–1-1-onto→((ω ↑o 2o) ↑o 𝑊))
47		f1oco 6867	. . . . . . 7 ⊢ ((𝐻:((ω ↑o 2o) ↑o 𝑊)–1-1-onto→(ω ↑o 𝑊) ∧ 𝐽:(ω ↑o (𝑊 ·o 2o))–1-1-onto→((ω ↑o 2o) ↑o 𝑊)) → (𝐻 ∘ 𝐽):(ω ↑o (𝑊 ·o 2o))–1-1-onto→(ω ↑o 𝑊))
48	24, 46, 47	syl2anc 582	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘ 𝐽):(ω ↑o (𝑊 ·o 2o))–1-1-onto→(ω ↑o 𝑊))
49		df-2o 8496	. . . . . . . . . . . 12 ⊢ 2o = suc 1o
50	49	oveq2i 7437	. . . . . . . . . . 11 ⊢ (𝑊 ·o 2o) = (𝑊 ·o suc 1o)
51		1on 8507	. . . . . . . . . . . 12 ⊢ 1o ∈ On
52		omsuc 8555	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ On ∧ 1o ∈ On) → (𝑊 ·o suc 1o) = ((𝑊 ·o 1o) +o 𝑊))
53	20, 51, 52	sylancl 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊 ·o suc 1o) = ((𝑊 ·o 1o) +o 𝑊))
54	50, 53	eqtrid 2780	. . . . . . . . . 10 ⊢ (𝜑 → (𝑊 ·o 2o) = ((𝑊 ·o 1o) +o 𝑊))
55		om1 8571	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ On → (𝑊 ·o 1o) = 𝑊)
56	20, 55	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊 ·o 1o) = 𝑊)
57	56	oveq1d 7441	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑊 ·o 1o) +o 𝑊) = (𝑊 +o 𝑊))
58	54, 57	eqtrd 2768	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ·o 2o) = (𝑊 +o 𝑊))
59	58	oveq2d 7442	. . . . . . . 8 ⊢ (𝜑 → (ω ↑o (𝑊 ·o 2o)) = (ω ↑o (𝑊 +o 𝑊)))
60		oeoa 8626	. . . . . . . . 9 ⊢ ((ω ∈ On ∧ 𝑊 ∈ On ∧ 𝑊 ∈ On) → (ω ↑o (𝑊 +o 𝑊)) = ((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
61	7, 20, 20, 60	mp3an2i 1462	. . . . . . . 8 ⊢ (𝜑 → (ω ↑o (𝑊 +o 𝑊)) = ((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
62	59, 61	eqtrd 2768	. . . . . . 7 ⊢ (𝜑 → (ω ↑o (𝑊 ·o 2o)) = ((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
63	62	f1oeq2d 6840	. . . . . 6 ⊢ (𝜑 → ((𝐻 ∘ 𝐽):(ω ↑o (𝑊 ·o 2o))–1-1-onto→(ω ↑o 𝑊) ↔ (𝐻 ∘ 𝐽):((ω ↑o 𝑊) ·o (ω ↑o 𝑊))–1-1-onto→(ω ↑o 𝑊)))
64	48, 63	mpbid 231	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ 𝐽):((ω ↑o 𝑊) ·o (ω ↑o 𝑊))–1-1-onto→(ω ↑o 𝑊))
65		oecl 8566	. . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑊 ∈ On) → (ω ↑o 𝑊) ∈ On)
66	8, 20, 65	syl2anc 582	. . . . . 6 ⊢ (𝜑 → (ω ↑o 𝑊) ∈ On)
67		infxpenc.z	. . . . . . 7 ⊢ 𝑍 = (𝑥 ∈ (ω ↑o 𝑊), 𝑦 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑥) +o 𝑦))
68	67	omxpenlem 9106	. . . . . 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)))
69	66, 66, 68	syl2anc 582	. . . . 5 ⊢ (𝜑 → 𝑍:((ω ↑o 𝑊) × (ω ↑o 𝑊))–1-1-onto→((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
70		f1oco 6867	. . . . 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))
71	64, 69, 70	syl2anc 582	. . . 4 ⊢ (𝜑 → ((𝐻 ∘ 𝐽) ∘ 𝑍):((ω ↑o 𝑊) × (ω ↑o 𝑊))–1-1-onto→(ω ↑o 𝑊))
72		f1of 6844	. . . . . . . . . 10 ⊢ (𝑁:𝐴–1-1-onto→(ω ↑o 𝑊) → 𝑁:𝐴⟶(ω ↑o 𝑊))
73	1, 72	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑁:𝐴⟶(ω ↑o 𝑊))
74	73	feqmptd 6972	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)))
75	74	f1oeq1d 6839	. . . . . . 7 ⊢ (𝜑 → (𝑁:𝐴–1-1-onto→(ω ↑o 𝑊) ↔ (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)):𝐴–1-1-onto→(ω ↑o 𝑊)))
76	1, 75	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)):𝐴–1-1-onto→(ω ↑o 𝑊))
77	73	feqmptd 6972	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)))
78	77	f1oeq1d 6839	. . . . . . 7 ⊢ (𝜑 → (𝑁:𝐴–1-1-onto→(ω ↑o 𝑊) ↔ (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)):𝐴–1-1-onto→(ω ↑o 𝑊)))
79	1, 78	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)):𝐴–1-1-onto→(ω ↑o 𝑊))
80	76, 79	xpf1o 9172	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩):(𝐴 × 𝐴)–1-1-onto→((ω ↑o 𝑊) × (ω ↑o 𝑊)))
81		infxpenc.t	. . . . . 6 ⊢ 𝑇 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩)
82		f1oeq1 6832	. . . . . 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))))
83	81, 82	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)))
84	80, 83	sylibr 233	. . . 4 ⊢ (𝜑 → 𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑o 𝑊) × (ω ↑o 𝑊)))
85		f1oco 6867	. . . 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))
86	71, 84, 85	syl2anc 582	. . 3 ⊢ (𝜑 → (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑o 𝑊))
87		f1oco 6867	. . 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)
88	3, 86, 87	syl2anc 582	. 2 ⊢ (𝜑 → (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto→𝐴)
89		infxpenc.g	. . 3 ⊢ 𝐺 = (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇))
90		f1oeq1 6832	. . 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))
91	89, 90	ax-mp 5	. 2 ⊢ (𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴 ↔ (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto→𝐴)
92	88, 91	sylibr 233	1 ⊢ (𝜑 → 𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴)

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098  {crab 3430   ∖ cdif 3946   ⊆ wss 3949  ∅c0 4326  ⟨cop 4638   class class class wbr 5152   ↦ cmpt 5235   I cid 5579   × cxp 5680  ^◡ccnv 5681   ↾ cres 5684   ∘ ccom 5686  Oncon0 6374  suc csuc 6376  ⟶wf 6549  –1-1-onto→wf1o 6552  ‘cfv 6553  (class class class)co 7426   ∈ cmpo 7428  ωcom 7878  1_oc1o 8488  2_oc2o 8489   +_o coa 8492   ·_o comu 8493   ↑_o coe 8494   ↑_m cmap 8853   finSupp cfsupp 9395   CNF ccnf 9694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-inf2 9674

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-supp 8174  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-seqom 8477  df-1o 8495  df-2o 8496  df-oadd 8499  df-omul 8500  df-oexp 8501  df-er 8733  df-map 8855  df-en 8973  df-dom 8974  df-sdom 8975  df-fin 8976  df-fsupp 9396  df-oi 9543  df-cnf 9695

This theorem is referenced by:  infxpenc2lem2  10053