Theorem infxpenc 10136

Description: A canonical version of infxpen 10132, by a completely different approach (although it uses infxpen 10132 via xpomen 10133). Using Cantor's normal form, we can show that 𝐴 ↑_o 𝐵 respects equinumerosity (oef1o 9736), 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 9744.) (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)⟶1-1ontoω)
infxpenc.5	⊢ (𝜑 → (𝐹‘∅) = ∅)
infxpenc.6	⊢ (𝜑 → 𝑁:𝐴⟶1-1onto(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)⟶1-1onto𝐴)

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⟶1-1onto(ω ↑o 𝑊))
2		f1ocnv 6867	. . . 4 ⊢ (𝑁:𝐴⟶1-1onto(ω ↑o 𝑊) → ◡𝑁:(ω ↑o 𝑊)⟶1-1onto𝐴)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → ◡𝑁:(ω ↑o 𝑊)⟶1-1onto𝐴)
4		infxpenc.4	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(ω ↑o 2o)⟶1-1ontoω)
5		f1oi 6894	. . . . . . . . 9 ⊢ ( I ↾ 𝑊):𝑊⟶1-1onto𝑊
6	5	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ( I ↾ 𝑊):𝑊⟶1-1onto𝑊)
7		omelon 9684	. . . . . . . . . . 11 ⊢ ω ∈ On
8	7	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ω ∈ On)
9		2on 8528	. . . . . . . . . 10 ⊢ 2o ∈ On
10		oecl 8583	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 2o ∈ On) → (ω ↑o 2o) ∈ On)
11	8, 9, 10	sylancl 598	. . . . . . . . 9 ⊢ (𝜑 → (ω ↑o 2o) ∈ On)
12	9	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 2o ∈ On)
13		peano1 7934	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
14	13	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∅ ∈ ω)
15		oen0 8633	. . . . . . . . . 10 ⊢ (((ω ∈ On ∧ 2o ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 2o))
16	8, 12, 14, 15	syl21anc 851	. . . . . . . . 9 ⊢ (𝜑 → ∅ ∈ (ω ↑o 2o))
17		ondif1 8547	. . . . . . . . 9 ⊢ ((ω ↑o 2o) ∈ (On ∖ 1o) ↔ ((ω ↑o 2o) ∈ On ∧ ∅ ∈ (ω ↑o 2o)))
18	11, 16, 17	sylanbrc 595	. . . . . . . 8 ⊢ (𝜑 → (ω ↑o 2o) ∈ (On ∖ 1o))
19		infxpenc.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1o))
20	19	eldifad 4054	. . . . . . . 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 9736	. . . . . . 7 ⊢ (𝜑 → 𝐻:((ω ↑o 2o) ↑o 𝑊)⟶1-1onto(ω ↑o 𝑊))
25		f1oi 6894	. . . . . . . . . 10 ⊢ ( I ↾ ω):ω⟶1-1ontoω
26	25	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ( I ↾ ω):ω⟶1-1ontoω)
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 9137	. . . . . . . . . 10 ⊢ ((𝑊 ∈ On ∧ 2o ∈ On) → (𝑌 ∘ ◡𝑋):(𝑊 ·o 2o)⟶1-1onto(2o ·o 𝑊))
30	20, 9, 29	sylancl 598	. . . . . . . . 9 ⊢ (𝜑 → (𝑌 ∘ ◡𝑋):(𝑊 ·o 2o)⟶1-1onto(2o ·o 𝑊))
31		ondif1 8547	. . . . . . . . . . 11 ⊢ (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
32	7, 13, 31	mpbir2an 724	. . . . . . . . . 10 ⊢ ω ∈ (On ∖ 1o)
33	32	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ω ∈ (On ∖ 1o))
34		omcl 8582	. . . . . . . . . 10 ⊢ ((𝑊 ∈ On ∧ 2o ∈ On) → (𝑊 ·o 2o) ∈ On)
35	20, 9, 34	sylancl 598	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ·o 2o) ∈ On)
36		omcl 8582	. . . . . . . . . 10 ⊢ ((2o ∈ On ∧ 𝑊 ∈ On) → (2o ·o 𝑊) ∈ On)
37	12, 20, 36	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (2o ·o 𝑊) ∈ On)
38		fvresi 7209	. . . . . . . . . 10 ⊢ (∅ ∈ ω → (( I ↾ ω)‘∅) = ∅)
39	13, 38	mp1i 14	. . . . . . . . 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 9736	. . . . . . . 8 ⊢ (𝜑 → 𝐽:(ω ↑o (𝑊 ·o 2o))⟶1-1onto(ω ↑o (2o ·o 𝑊)))
43		oeoe 8646	. . . . . . . . . 10 ⊢ ((ω ∈ On ∧ 2o ∈ On ∧ 𝑊 ∈ On) → ((ω ↑o 2o) ↑o 𝑊) = (ω ↑o (2o ·o 𝑊)))
44	7, 12, 20, 43	mp3an2i 1496	. . . . . . . . 9 ⊢ (𝜑 → ((ω ↑o 2o) ↑o 𝑊) = (ω ↑o (2o ·o 𝑊)))
45	44	f1oeq3d 6851	. . . . . . . 8 ⊢ (𝜑 → (𝐽:(ω ↑o (𝑊 ·o 2o))⟶1-1onto((ω ↑o 2o) ↑o 𝑊) ↔ 𝐽:(ω ↑o (𝑊 ·o 2o))⟶1-1onto(ω ↑o (2o ·o 𝑊))))
46	42, 45	mpbird 260	. . . . . . 7 ⊢ (𝜑 → 𝐽:(ω ↑o (𝑊 ·o 2o))⟶1-1onto((ω ↑o 2o) ↑o 𝑊))
47		f1oco 6878	. . . . . . 7 ⊢ ((𝐻:((ω ↑o 2o) ↑o 𝑊)⟶1-1onto(ω ↑o 𝑊) ∧ 𝐽:(ω ↑o (𝑊 ·o 2o))⟶1-1onto((ω ↑o 2o) ↑o 𝑊)) → (𝐻 ∘ 𝐽):(ω ↑o (𝑊 ·o 2o))⟶1-1onto(ω ↑o 𝑊))
48	24, 46, 47	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐻 ∘ 𝐽):(ω ↑o (𝑊 ·o 2o))⟶1-1onto(ω ↑o 𝑊))
49		df-2o 8515	. . . . . . . . . . . 12 ⊢ 2o = suc 1o
50	49	oveq2i 7462	. . . . . . . . . . 11 ⊢ (𝑊 ·o 2o) = (𝑊 ·o suc 1o)
51		1on 8527	. . . . . . . . . . . 12 ⊢ 1o ∈ On
52		omsuc 8572	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ On ∧ 1o ∈ On) → (𝑊 ·o suc 1o) = ((𝑊 ·o 1o) +o 𝑊))
53	20, 51, 52	sylancl 598	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊 ·o suc 1o) = ((𝑊 ·o 1o) +o 𝑊))
54	50, 53	eqtrid 2810	. . . . . . . . . 10 ⊢ (𝜑 → (𝑊 ·o 2o) = ((𝑊 ·o 1o) +o 𝑊))
55		om1 8588	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ On → (𝑊 ·o 1o) = 𝑊)
56	20, 55	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑊 ·o 1o) = 𝑊)
57	56	oveq1d 7466	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑊 ·o 1o) +o 𝑊) = (𝑊 +o 𝑊))
58	54, 57	eqtrd 2798	. . . . . . . . 9 ⊢ (𝜑 → (𝑊 ·o 2o) = (𝑊 +o 𝑊))
59	58	oveq2d 7467	. . . . . . . 8 ⊢ (𝜑 → (ω ↑o (𝑊 ·o 2o)) = (ω ↑o (𝑊 +o 𝑊)))
60		oeoa 8644	. . . . . . . . 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 1496	. . . . . . . 8 ⊢ (𝜑 → (ω ↑o (𝑊 +o 𝑊)) = ((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
62	59, 61	eqtrd 2798	. . . . . . 7 ⊢ (𝜑 → (ω ↑o (𝑊 ·o 2o)) = ((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
63	62	f1oeq2d 6850	. . . . . 6 ⊢ (𝜑 → ((𝐻 ∘ 𝐽):(ω ↑o (𝑊 ·o 2o))⟶1-1onto(ω ↑o 𝑊) ↔ (𝐻 ∘ 𝐽):((ω ↑o 𝑊) ·o (ω ↑o 𝑊))⟶1-1onto(ω ↑o 𝑊)))
64	48, 63	mpbid 235	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ 𝐽):((ω ↑o 𝑊) ·o (ω ↑o 𝑊))⟶1-1onto(ω ↑o 𝑊))
65		oecl 8583	. . . . . . 7 ⊢ ((ω ∈ On ∧ 𝑊 ∈ On) → (ω ↑o 𝑊) ∈ On)
66	8, 20, 65	syl2anc 596	. . . . . 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 9135	. . . . . 6 ⊢ (((ω ↑o 𝑊) ∈ On ∧ (ω ↑o 𝑊) ∈ On) → 𝑍:((ω ↑o 𝑊) × (ω ↑o 𝑊))⟶1-1onto((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
69	66, 66, 68	syl2anc 596	. . . . 5 ⊢ (𝜑 → 𝑍:((ω ↑o 𝑊) × (ω ↑o 𝑊))⟶1-1onto((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
70		f1oco 6878	. . . . 5 ⊢ (((𝐻 ∘ 𝐽):((ω ↑o 𝑊) ·o (ω ↑o 𝑊))⟶1-1onto(ω ↑o 𝑊) ∧ 𝑍:((ω ↑o 𝑊) × (ω ↑o 𝑊))⟶1-1onto((ω ↑o 𝑊) ·o (ω ↑o 𝑊))) → ((𝐻 ∘ 𝐽) ∘ 𝑍):((ω ↑o 𝑊) × (ω ↑o 𝑊))⟶1-1onto(ω ↑o 𝑊))
71	64, 69, 70	syl2anc 596	. . . 4 ⊢ (𝜑 → ((𝐻 ∘ 𝐽) ∘ 𝑍):((ω ↑o 𝑊) × (ω ↑o 𝑊))⟶1-1onto(ω ↑o 𝑊))
72		f1of 6854	. . . . . . . . . 10 ⊢ (𝑁:𝐴⟶1-1onto(ω ↑o 𝑊) → 𝑁:𝐴⟶(ω ↑o 𝑊))
73	1, 72	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑁:𝐴⟶(ω ↑o 𝑊))
74	73	feqmptd 6984	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)))
75	74	f1oeq1d 6849	. . . . . . 7 ⊢ (𝜑 → (𝑁:𝐴⟶1-1onto(ω ↑o 𝑊) ↔ (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)):𝐴⟶1-1onto(ω ↑o 𝑊)))
76	1, 75	mpbid 235	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (𝑁‘𝑥)):𝐴⟶1-1onto(ω ↑o 𝑊))
77	73	feqmptd 6984	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)))
78	77	f1oeq1d 6849	. . . . . . 7 ⊢ (𝜑 → (𝑁:𝐴⟶1-1onto(ω ↑o 𝑊) ↔ (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)):𝐴⟶1-1onto(ω ↑o 𝑊)))
79	1, 78	mpbid 235	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (𝑁‘𝑦)):𝐴⟶1-1onto(ω ↑o 𝑊))
80	76, 79	xpf1o 9196	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩):(𝐴 × 𝐴)⟶1-1onto((ω ↑o 𝑊) × (ω ↑o 𝑊)))
81		infxpenc.t	. . . . . 6 ⊢ 𝑇 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩)
82		f1oeq1 6842	. . . . . 6 ⊢ (𝑇 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩) → (𝑇:(𝐴 × 𝐴)⟶1-1onto((ω ↑o 𝑊) × (ω ↑o 𝑊)) ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩):(𝐴 × 𝐴)⟶1-1onto((ω ↑o 𝑊) × (ω ↑o 𝑊))))
83	81, 82	ax-mp 5	. . . . 5 ⊢ (𝑇:(𝐴 × 𝐴)⟶1-1onto((ω ↑o 𝑊) × (ω ↑o 𝑊)) ↔ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ ⟨(𝑁‘𝑥), (𝑁‘𝑦)⟩):(𝐴 × 𝐴)⟶1-1onto((ω ↑o 𝑊) × (ω ↑o 𝑊)))
84	80, 83	sylibr 237	. . . 4 ⊢ (𝜑 → 𝑇:(𝐴 × 𝐴)⟶1-1onto((ω ↑o 𝑊) × (ω ↑o 𝑊)))
85		f1oco 6878	. . . 4 ⊢ ((((𝐻 ∘ 𝐽) ∘ 𝑍):((ω ↑o 𝑊) × (ω ↑o 𝑊))⟶1-1onto(ω ↑o 𝑊) ∧ 𝑇:(𝐴 × 𝐴)⟶1-1onto((ω ↑o 𝑊) × (ω ↑o 𝑊))) → (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)⟶1-1onto(ω ↑o 𝑊))
86	71, 84, 85	syl2anc 596	. . 3 ⊢ (𝜑 → (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)⟶1-1onto(ω ↑o 𝑊))
87		f1oco 6878	. . 3 ⊢ ((◡𝑁:(ω ↑o 𝑊)⟶1-1onto𝐴 ∧ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)⟶1-1onto(ω ↑o 𝑊)) → (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)⟶1-1onto𝐴)
88	3, 86, 87	syl2anc 596	. 2 ⊢ (𝜑 → (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)⟶1-1onto𝐴)
89		infxpenc.g	. . 3 ⊢ 𝐺 = (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇))
90		f1oeq1 6842	. . 3 ⊢ (𝐺 = (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)) → (𝐺:(𝐴 × 𝐴)⟶1-1onto𝐴 ↔ (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)⟶1-1onto𝐴))
91	89, 90	ax-mp 5	. 2 ⊢ (𝐺:(𝐴 × 𝐴)⟶1-1onto𝐴 ↔ (◡𝑁 ∘ (((𝐻 ∘ 𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)⟶1-1onto𝐴)
92	88, 91	sylibr 237	1 ⊢ (𝜑 → 𝐺:(𝐴 × 𝐴)⟶1-1onto𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147  {crab 3416   ⊆ wss 3900   ∖ cdif 4048  ∅c0 4281  ⟨cop 4840   class class class wbr 5108   ↦ cmpt 5192   I cid 5552   × cxp 5656  ^◡ccnv 5657   ↾ cres 5660   ∘ ccom 5662  Oncon0 6388  suc csuc 6390  ⟶wf 6562  ⟶1-1ontowf1o 6565  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  ωcom 7911  1_oc1o 8507  2_oc2o 8508   +_o coa 8511   ·_o comu 8512   ↑_o coe 8513   ↑_m cmap 8885   finSupp cfsupp 9391   CNF ccnf 9699

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-inf2 9679

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-isom 6575  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-supp 8215  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-seqom 8496  df-1o 8514  df-2o 8515  df-oadd 8518  df-omul 8519  df-oexp 8520  df-er 8755  df-map 8887  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-fsupp 9392  df-oi 9541  df-cnf 9700

This theorem is used by:  infxpenc2lem2  10138