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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 𝐴𝑜 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 (𝜑𝐴 ∈ On)
infxpenc.2 (𝜑 → ω ⊆ 𝐴)
infxpenc.3 (𝜑𝑊 ∈ (On ∖ 1𝑜))
infxpenc.4 (𝜑𝐹:(ω ↑𝑜 2𝑜)–1-1-onto→ω)
infxpenc.5 (𝜑 → (𝐹‘∅) = ∅)
infxpenc.6 (𝜑𝑁:𝐴1-1-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(𝑌𝑋))))
infxpenc.x 𝑋 = (𝑧 ∈ 2𝑜, 𝑤𝑊 ↦ ((𝑊 ·𝑜 𝑧) +𝑜 𝑤))
infxpenc.y 𝑌 = (𝑧 ∈ 2𝑜, 𝑤𝑊 ↦ ((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 = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩)
infxpenc.g 𝐺 = (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇))
Assertion
Ref Expression
infxpenc (𝜑𝐺:(𝐴 × 𝐴)–1-1-onto𝐴)
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
StepHypRef Expression
1 infxpenc.6 . . . 4 (𝜑𝑁:𝐴1-1-onto→(ω ↑𝑜 𝑊))
2 f1ocnv 6368 . . . 4 (𝑁:𝐴1-1-onto→(ω ↑𝑜 𝑊) → 𝑁:(ω ↑𝑜 𝑊)–1-1-onto𝐴)
31, 2syl 17 . . 3 (𝜑𝑁:(ω ↑𝑜 𝑊)–1-1-onto𝐴)
4 infxpenc.4 . . . . . . . 8 (𝜑𝐹:(ω ↑𝑜 2𝑜)–1-1-onto→ω)
5 f1oi 6393 . . . . . . . . 9 ( I ↾ 𝑊):𝑊1-1-onto𝑊
65a1i 11 . . . . . . . 8 (𝜑 → ( I ↾ 𝑊):𝑊1-1-onto𝑊)
7 omelon 8793 . . . . . . . . . . 11 ω ∈ On
87a1i 11 . . . . . . . . . 10 (𝜑 → ω ∈ On)
9 2on 7808 . . . . . . . . . 10 2𝑜 ∈ On
10 oecl 7857 . . . . . . . . . 10 ((ω ∈ On ∧ 2𝑜 ∈ On) → (ω ↑𝑜 2𝑜) ∈ On)
118, 9, 10sylancl 581 . . . . . . . . 9 (𝜑 → (ω ↑𝑜 2𝑜) ∈ On)
129a1i 11 . . . . . . . . . 10 (𝜑 → 2𝑜 ∈ On)
13 peano1 7319 . . . . . . . . . . 11 ∅ ∈ ω
1413a1i 11 . . . . . . . . . 10 (𝜑 → ∅ ∈ ω)
15 oen0 7906 . . . . . . . . . 10 (((ω ∈ On ∧ 2𝑜 ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑𝑜 2𝑜))
168, 12, 14, 15syl21anc 867 . . . . . . . . 9 (𝜑 → ∅ ∈ (ω ↑𝑜 2𝑜))
17 ondif1 7821 . . . . . . . . 9 ((ω ↑𝑜 2𝑜) ∈ (On ∖ 1𝑜) ↔ ((ω ↑𝑜 2𝑜) ∈ On ∧ ∅ ∈ (ω ↑𝑜 2𝑜)))
1811, 16, 17sylanbrc 579 . . . . . . . 8 (𝜑 → (ω ↑𝑜 2𝑜) ∈ (On ∖ 1𝑜))
19 infxpenc.3 . . . . . . . . 9 (𝜑𝑊 ∈ (On ∖ 1𝑜))
2019eldifad 3781 . . . . . . . 8 (𝜑𝑊 ∈ 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))
244, 6, 18, 20, 8, 20, 21, 22, 23oef1o 8845 . . . . . . 7 (𝜑𝐻:((ω ↑𝑜 2𝑜) ↑𝑜 𝑊)–1-1-onto→(ω ↑𝑜 𝑊))
25 f1oi 6393 . . . . . . . . . 10 ( I ↾ ω):ω–1-1-onto→ω
2625a1i 11 . . . . . . . . 9 (𝜑 → ( I ↾ ω):ω–1-1-onto→ω)
27 infxpenc.x . . . . . . . . . . 11 𝑋 = (𝑧 ∈ 2𝑜, 𝑤𝑊 ↦ ((𝑊 ·𝑜 𝑧) +𝑜 𝑤))
28 infxpenc.y . . . . . . . . . . 11 𝑌 = (𝑧 ∈ 2𝑜, 𝑤𝑊 ↦ ((2𝑜 ·𝑜 𝑤) +𝑜 𝑧))
2927, 28omf1o 8305 . . . . . . . . . 10 ((𝑊 ∈ On ∧ 2𝑜 ∈ On) → (𝑌𝑋):(𝑊 ·𝑜 2𝑜)–1-1-onto→(2𝑜 ·𝑜 𝑊))
3020, 9, 29sylancl 581 . . . . . . . . 9 (𝜑 → (𝑌𝑋):(𝑊 ·𝑜 2𝑜)–1-1-onto→(2𝑜 ·𝑜 𝑊))
31 ondif1 7821 . . . . . . . . . . 11 (ω ∈ (On ∖ 1𝑜) ↔ (ω ∈ On ∧ ∅ ∈ ω))
327, 13, 31mpbir2an 703 . . . . . . . . . 10 ω ∈ (On ∖ 1𝑜)
3332a1i 11 . . . . . . . . 9 (𝜑 → ω ∈ (On ∖ 1𝑜))
34 omcl 7856 . . . . . . . . . 10 ((𝑊 ∈ On ∧ 2𝑜 ∈ On) → (𝑊 ·𝑜 2𝑜) ∈ On)
3520, 9, 34sylancl 581 . . . . . . . . 9 (𝜑 → (𝑊 ·𝑜 2𝑜) ∈ On)
36 omcl 7856 . . . . . . . . . 10 ((2𝑜 ∈ On ∧ 𝑊 ∈ On) → (2𝑜 ·𝑜 𝑊) ∈ On)
3712, 20, 36syl2anc 580 . . . . . . . . 9 (𝜑 → (2𝑜 ·𝑜 𝑊) ∈ On)
38 fvresi 6668 . . . . . . . . . 10 (∅ ∈ ω → (( I ↾ ω)‘∅) = ∅)
3913, 38mp1i 13 . . . . . . . . 9 (𝜑 → (( I ↾ ω)‘∅) = ∅)
40 infxpenc.l . . . . . . . . 9 𝐿 = (𝑦 ∈ {𝑥 ∈ (ω ↑𝑚 (𝑊 ·𝑜 2𝑜)) ∣ 𝑥 finSupp ∅} ↦ (( I ↾ ω) ∘ (𝑦(𝑌𝑋))))
41 infxpenc.j . . . . . . . . 9 𝐽 = (((ω CNF (2𝑜 ·𝑜 𝑊)) ∘ 𝐿) ∘ (ω CNF (𝑊 ·𝑜 2𝑜)))
4226, 30, 33, 35, 8, 37, 39, 40, 41oef1o 8845 . . . . . . . 8 (𝜑𝐽:(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 (2𝑜 ·𝑜 𝑊)))
43 oeoe 7919 . . . . . . . . . 10 ((ω ∈ On ∧ 2𝑜 ∈ On ∧ 𝑊 ∈ On) → ((ω ↑𝑜 2𝑜) ↑𝑜 𝑊) = (ω ↑𝑜 (2𝑜 ·𝑜 𝑊)))
448, 12, 20, 43syl3anc 1491 . . . . . . . . 9 (𝜑 → ((ω ↑𝑜 2𝑜) ↑𝑜 𝑊) = (ω ↑𝑜 (2𝑜 ·𝑜 𝑊)))
45 f1oeq3 6347 . . . . . . . . 9 (((ω ↑𝑜 2𝑜) ↑𝑜 𝑊) = (ω ↑𝑜 (2𝑜 ·𝑜 𝑊)) → (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))))
4644, 45syl 17 . . . . . . . 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))))
4742, 46mpbird 249 . . . . . . 7 (𝜑𝐽:(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→((ω ↑𝑜 2𝑜) ↑𝑜 𝑊))
48 f1oco 6378 . . . . . . 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)) → (𝐻𝐽):(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 𝑊))
4924, 47, 48syl2anc 580 . . . . . 6 (𝜑 → (𝐻𝐽):(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 𝑊))
50 df-2o 7800 . . . . . . . . . . . 12 2𝑜 = suc 1𝑜
5150oveq2i 6889 . . . . . . . . . . 11 (𝑊 ·𝑜 2𝑜) = (𝑊 ·𝑜 suc 1𝑜)
52 1on 7806 . . . . . . . . . . . 12 1𝑜 ∈ On
53 omsuc 7846 . . . . . . . . . . . 12 ((𝑊 ∈ On ∧ 1𝑜 ∈ On) → (𝑊 ·𝑜 suc 1𝑜) = ((𝑊 ·𝑜 1𝑜) +𝑜 𝑊))
5420, 52, 53sylancl 581 . . . . . . . . . . 11 (𝜑 → (𝑊 ·𝑜 suc 1𝑜) = ((𝑊 ·𝑜 1𝑜) +𝑜 𝑊))
5551, 54syl5eq 2845 . . . . . . . . . 10 (𝜑 → (𝑊 ·𝑜 2𝑜) = ((𝑊 ·𝑜 1𝑜) +𝑜 𝑊))
56 om1 7862 . . . . . . . . . . . 12 (𝑊 ∈ On → (𝑊 ·𝑜 1𝑜) = 𝑊)
5720, 56syl 17 . . . . . . . . . . 11 (𝜑 → (𝑊 ·𝑜 1𝑜) = 𝑊)
5857oveq1d 6893 . . . . . . . . . 10 (𝜑 → ((𝑊 ·𝑜 1𝑜) +𝑜 𝑊) = (𝑊 +𝑜 𝑊))
5955, 58eqtrd 2833 . . . . . . . . 9 (𝜑 → (𝑊 ·𝑜 2𝑜) = (𝑊 +𝑜 𝑊))
6059oveq2d 6894 . . . . . . . 8 (𝜑 → (ω ↑𝑜 (𝑊 ·𝑜 2𝑜)) = (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)))
628, 20, 20, 61syl3anc 1491 . . . . . . . 8 (𝜑 → (ω ↑𝑜 (𝑊 +𝑜 𝑊)) = ((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
6360, 62eqtrd 2833 . . . . . . 7 (𝜑 → (ω ↑𝑜 (𝑊 ·𝑜 2𝑜)) = ((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
6463f1oeq2d 6352 . . . . . 6 (𝜑 → ((𝐻𝐽):(ω ↑𝑜 (𝑊 ·𝑜 2𝑜))–1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝐻𝐽):((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊)))
6549, 64mpbid 224 . . . . 5 (𝜑 → (𝐻𝐽):((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)
678, 20, 66syl2anc 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))
6968omxpenlem 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)))
7067, 67, 69syl2anc 580 . . . . 5 (𝜑𝑍:((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))–1-1-onto→((ω ↑𝑜 𝑊) ·𝑜 (ω ↑𝑜 𝑊)))
71 f1oco 6378 . . . . 5 (((𝐻𝐽):((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))) → ((𝐻𝐽) ∘ 𝑍):((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊))
7265, 70, 71syl2anc 580 . . . 4 (𝜑 → ((𝐻𝐽) ∘ 𝑍):((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))–1-1-onto→(ω ↑𝑜 𝑊))
73 f1of 6356 . . . . . . . . . 10 (𝑁:𝐴1-1-onto→(ω ↑𝑜 𝑊) → 𝑁:𝐴⟶(ω ↑𝑜 𝑊))
741, 73syl 17 . . . . . . . . 9 (𝜑𝑁:𝐴⟶(ω ↑𝑜 𝑊))
7574feqmptd 6474 . . . . . . . 8 (𝜑𝑁 = (𝑥𝐴 ↦ (𝑁𝑥)))
76 f1oeq1 6345 . . . . . . . 8 (𝑁 = (𝑥𝐴 ↦ (𝑁𝑥)) → (𝑁:𝐴1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑥𝐴 ↦ (𝑁𝑥)):𝐴1-1-onto→(ω ↑𝑜 𝑊)))
7775, 76syl 17 . . . . . . 7 (𝜑 → (𝑁:𝐴1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑥𝐴 ↦ (𝑁𝑥)):𝐴1-1-onto→(ω ↑𝑜 𝑊)))
781, 77mpbid 224 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (𝑁𝑥)):𝐴1-1-onto→(ω ↑𝑜 𝑊))
7974feqmptd 6474 . . . . . . . 8 (𝜑𝑁 = (𝑦𝐴 ↦ (𝑁𝑦)))
80 f1oeq1 6345 . . . . . . . 8 (𝑁 = (𝑦𝐴 ↦ (𝑁𝑦)) → (𝑁:𝐴1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑦𝐴 ↦ (𝑁𝑦)):𝐴1-1-onto→(ω ↑𝑜 𝑊)))
8179, 80syl 17 . . . . . . 7 (𝜑 → (𝑁:𝐴1-1-onto→(ω ↑𝑜 𝑊) ↔ (𝑦𝐴 ↦ (𝑁𝑦)):𝐴1-1-onto→(ω ↑𝑜 𝑊)))
821, 81mpbid 224 . . . . . 6 (𝜑 → (𝑦𝐴 ↦ (𝑁𝑦)):𝐴1-1-onto→(ω ↑𝑜 𝑊))
8378, 82xpf1o 8364 . . . . 5 (𝜑 → (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩):(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊)))
84 infxpenc.t . . . . . 6 𝑇 = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩)
85 f1oeq1 6345 . . . . . 6 (𝑇 = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩) → (𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊)) ↔ (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩):(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊))))
8684, 85ax-mp 5 . . . . 5 (𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊)) ↔ (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩):(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊)))
8783, 86sylibr 226 . . . 4 (𝜑𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑𝑜 𝑊) × (ω ↑𝑜 𝑊)))
88 f1oco 6378 . . . 4 ((((𝐻𝐽) ∘ 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))) → (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑𝑜 𝑊))
8972, 87, 88syl2anc 580 . . 3 (𝜑 → (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑𝑜 𝑊))
90 f1oco 6378 . . 3 ((𝑁:(ω ↑𝑜 𝑊)–1-1-onto𝐴 ∧ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑𝑜 𝑊)) → (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto𝐴)
913, 89, 90syl2anc 580 . 2 (𝜑 → (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto𝐴)
92 infxpenc.g . . 3 𝐺 = (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇))
93 f1oeq1 6345 . . 3 (𝐺 = (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)) → (𝐺:(𝐴 × 𝐴)–1-1-onto𝐴 ↔ (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto𝐴))
9492, 93ax-mp 5 . 2 (𝐺:(𝐴 × 𝐴)–1-1-onto𝐴 ↔ (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto𝐴)
9591, 94sylibr 226 1 (𝜑𝐺:(𝐴 × 𝐴)–1-1-onto𝐴)
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-ontowf1o 6100  cfv 6101  (class class class)co 6878  cmpt2 6880  ωcom 7299  1𝑜c1o 7792  2𝑜c2o 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
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