Theorem infxpenc 10032

Description: A canonical version of infxpen 10028, by a completely different approach (although it uses infxpen 10028 via xpomen 10029). Using Cantor's normal form, we can show that 𝐴 ↑_o 𝐵 respects equinumerosity (oef1o 9712), 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 9720.) (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 6830	. . . 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 6856	. . . . . . . . 9 ⊢ ( I ↾ 𝑊):𝑊–1-1-onto→𝑊
6	5	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ( I ↾ 𝑊):𝑊–1-1-onto→𝑊)
7		omelon 9660	. . . . . . . . . . 11 ⊢ ω ∈ On
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ω ∈ On)
9		2on 8494	. . . . . . . . . 10 ⊢ 2o ∈ On
10		oecl 8549	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 2o ∈ On) → (ω ↑o 2o) ∈ On)
11	8, 9, 10	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → (ω ↑o 2o) ∈ On)
12	9	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2o ∈ On)
13		peano1 7884	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
14	13	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ ω)
15		oen0 8598	. . . . . . . . . 10 ⊢ (((ω ∈ On ∧ 2o ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 2o))
16	8, 12, 14, 15	syl21anc 837	. . . . . . . . 9 ⊢ (𝜑 → ∅ ∈ (ω ↑o 2o))
17		ondif1 8513	. . . . . . . . 9 ⊢ ((ω ↑o 2o) ∈ (On ∖ 1o) ↔ ((ω ↑o 2o) ∈ On ∧ ∅ ∈ (ω ↑o 2o)))
18	11, 16, 17	sylanbrc 583	. . . . . . . 8 ⊢ (𝜑 → (ω ↑o 2o) ∈ (On ∖ 1o))
19		infxpenc.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1o))
20	19	eldifad 3938	. . . . . . . 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 9712	. . . . . . 7 ⊢ (𝜑 → 𝐻:((ω ↑o 2o) ↑o 𝑊)–1-1-onto→(ω ↑o 𝑊))
25		f1oi 6856	. . . . . . . . . 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 9089	. . . . . . . . . 10 ⊢ ((𝑊 ∈ On ∧ 2o ∈ On) → (𝑌 ∘ ◡𝑋):(𝑊 ·o 2o)–1-1-onto→(2o ·o 𝑊))
30	20, 9, 29	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∘ ◡𝑋):(𝑊 ·o 2o)–1-1-onto→(2o ·o 𝑊))
31		ondif1 8513	. . . . . . . . . . 11 ⊢ (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
32	7, 13, 31	mpbir2an 711	. . . . . . . . . 10 ⊢ ω ∈ (On ∖ 1o)
33	32	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ω ∈ (On ∖ 1o))
34		omcl 8548	. . . . . . . . . 10 ⊢ ((𝑊 ∈ On ∧ 2o ∈ On) → (𝑊 ·o 2o) ∈ On)
35	20, 9, 34	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ·o 2o) ∈ On)
36		omcl 8548	. . . . . . . . . 10 ⊢ ((2o ∈ On ∧ 𝑊 ∈ On) → (2o ·o 𝑊) ∈ On)
37	12, 20, 36	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (2o ·o 𝑊) ∈ On)
38		fvresi 7165	. . . . . . . . . 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 9712	. . . . . . . 8 ⊢ (𝜑 → 𝐽:(ω ↑o (𝑊 ·o 2o))–1-1-onto→(ω ↑o (2o ·o 𝑊)))
43		oeoe 8611	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 2o ∈ On ∧ 𝑊 ∈ On) → ((ω ↑o 2o) ↑o 𝑊) = (ω ↑o (2o ·o 𝑊)))
44	7, 12, 20, 43	mp3an2i 1468	. . . . . . . . 9 ⊢ (𝜑 → ((ω ↑o 2o) ↑o 𝑊) = (ω ↑o (2o ·o 𝑊)))
45	44	f1oeq3d 6815	. . . . . . . 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 257	. . . . . . 7 ⊢ (𝜑 → 𝐽:(ω ↑o (𝑊 ·o 2o))–1-1-onto→((ω ↑o 2o) ↑o 𝑊))
47		f1oco 6841	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘ 𝐽):(ω ↑o (𝑊 ·o 2o))–1-1-onto→(ω ↑o 𝑊))
49		df-2o 8481	. . . . . . . . . . . 12 ⊢ 2o = suc 1o
50	49	oveq2i 7416	. . . . . . . . . . 11 ⊢ (𝑊 ·o 2o) = (𝑊 ·o suc 1o)
51		1on 8492	. . . . . . . . . . . 12 ⊢ 1o ∈ On
52		omsuc 8538	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ On ∧ 1o ∈ On) → (𝑊 ·o suc 1o) = ((𝑊 ·o 1o) +o 𝑊))
53	20, 51, 52	sylancl 586	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊 ·o suc 1o) = ((𝑊 ·o 1o) +o 𝑊))
54	50, 53	eqtrid 2782	. . . . . . . . . 10 ⊢ (𝜑 → (𝑊 ·o 2o) = ((𝑊 ·o 1o) +o 𝑊))
55		om1 8554	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ On → (𝑊 ·o 1o) = 𝑊)
56	20, 55	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊 ·o 1o) = 𝑊)
57	56	oveq1d 7420	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑊 ·o 1o) +o 𝑊) = (𝑊 +o 𝑊))
58	54, 57	eqtrd 2770	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ·o 2o) = (𝑊 +o 𝑊))
59	58	oveq2d 7421	. . . . . . . 8 ⊢ (𝜑 → (ω ↑o (𝑊 ·o 2o)) = (ω ↑o (𝑊 +o 𝑊)))
60		oeoa 8609	. . . . . . . . 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 1468	. . . . . . . 8 ⊢ (𝜑 → (ω ↑o (𝑊 +o 𝑊)) = ((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
62	59, 61	eqtrd 2770	. . . . . . 7 ⊢ (𝜑 → (ω ↑o (𝑊 ·o 2o)) = ((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
63	62	f1oeq2d 6814	. . . . . 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 232	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ 𝐽):((ω ↑o 𝑊) ·o (ω ↑o 𝑊))–1-1-onto→(ω ↑o 𝑊))
65		oecl 8549	. . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑊 ∈ On) → (ω ↑o 𝑊) ∈ On)
66	8, 20, 65	syl2anc 584	. . . . . 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 9087	. . . . . 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 584	. . . . 5 ⊢ (𝜑 → 𝑍:((ω ↑o 𝑊) × (ω ↑o 𝑊))–1-1-onto→((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
70		f1oco 6841	. . . . 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 584	. . . 4 ⊢ (𝜑 → ((𝐻 ∘ 𝐽) ∘ 𝑍):((ω ↑o 𝑊) × (ω ↑o 𝑊))–1-1-onto→(ω ↑o 𝑊))
72		f1of 6818	. . . . . . . . . 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 6947	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)))
75	74	f1oeq1d 6813	. . . . . . 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 232	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)):𝐴–1-1-onto→(ω ↑o 𝑊))
77	73	feqmptd 6947	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)))
78	77	f1oeq1d 6813	. . . . . . 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 232	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)):𝐴–1-1-onto→(ω ↑o 𝑊))
80	76, 79	xpf1o 9153	. . . . 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 6806	. . . . . 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 234	. . . 4 ⊢ (𝜑 → 𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑o 𝑊) × (ω ↑o 𝑊)))
85		f1oco 6841	. . . 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 584	. . 3 ⊢ (𝜑 → (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑o 𝑊))
87		f1oco 6841	. . 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 584	. 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 6806	. . 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 234	1 ⊢ (𝜑 → 𝐺:(𝐴 × 𝐴)–1-1-onto→𝐴)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108  {crab 3415   ∖ cdif 3923   ⊆ wss 3926  ∅c0 4308  ⟨cop 4607   class class class wbr 5119   ↦ cmpt 5201   I cid 5547   × cxp 5652  ^◡ccnv 5653   ↾ cres 5656   ∘ ccom 5658  Oncon0 6352  suc csuc 6354  ⟶wf 6527  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  ωcom 7861  1_oc1o 8473  2_oc2o 8474   +_o coa 8477   ·_o comu 8478   ↑_o coe 8479   ↑_m cmap 8840   finSupp cfsupp 9373   CNF ccnf 9675

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-supp 8160  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-seqom 8462  df-1o 8480  df-2o 8481  df-oadd 8484  df-omul 8485  df-oexp 8486  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fsupp 9374  df-oi 9524  df-cnf 9676

This theorem is referenced by:  infxpenc2lem2  10034