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Theorem infxpenc 10056
Description: A canonical version of infxpen 10052, by a completely different approach (although it uses infxpen 10052 via xpomen 10053). Using Cantor's normal form, we can show that 𝐴o 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 (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 6861 . . . 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 6887 . . . . . . . . 9 ( I ↾ 𝑊):𝑊1-1-onto𝑊
65a1i 11 . . . . . . . 8 (𝜑 → ( I ↾ 𝑊):𝑊1-1-onto𝑊)
7 omelon 9684 . . . . . . . . . . 11 ω ∈ On
87a1i 11 . . . . . . . . . 10 (𝜑 → ω ∈ On)
9 2on 8519 . . . . . . . . . 10 2o ∈ On
10 oecl 8574 . . . . . . . . . 10 ((ω ∈ On ∧ 2o ∈ On) → (ω ↑o 2o) ∈ On)
118, 9, 10sylancl 586 . . . . . . . . 9 (𝜑 → (ω ↑o 2o) ∈ On)
129a1i 11 . . . . . . . . . 10 (𝜑 → 2o ∈ On)
13 peano1 7911 . . . . . . . . . . 11 ∅ ∈ ω
1413a1i 11 . . . . . . . . . 10 (𝜑 → ∅ ∈ ω)
15 oen0 8623 . . . . . . . . . 10 (((ω ∈ On ∧ 2o ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 2o))
168, 12, 14, 15syl21anc 838 . . . . . . . . 9 (𝜑 → ∅ ∈ (ω ↑o 2o))
17 ondif1 8538 . . . . . . . . 9 ((ω ↑o 2o) ∈ (On ∖ 1o) ↔ ((ω ↑o 2o) ∈ On ∧ ∅ ∈ (ω ↑o 2o)))
1811, 16, 17sylanbrc 583 . . . . . . . 8 (𝜑 → (ω ↑o 2o) ∈ (On ∖ 1o))
19 infxpenc.3 . . . . . . . . 9 (𝜑𝑊 ∈ (On ∖ 1o))
2019eldifad 3975 . . . . . . . 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 9736 . . . . . . 7 (𝜑𝐻:((ω ↑o 2o) ↑o 𝑊)–1-1-onto→(ω ↑o 𝑊))
25 f1oi 6887 . . . . . . . . . 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 9114 . . . . . . . . . 10 ((𝑊 ∈ On ∧ 2o ∈ On) → (𝑌𝑋):(𝑊 ·o 2o)–1-1-onto→(2o ·o 𝑊))
3020, 9, 29sylancl 586 . . . . . . . . 9 (𝜑 → (𝑌𝑋):(𝑊 ·o 2o)–1-1-onto→(2o ·o 𝑊))
31 ondif1 8538 . . . . . . . . . . 11 (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
327, 13, 31mpbir2an 711 . . . . . . . . . 10 ω ∈ (On ∖ 1o)
3332a1i 11 . . . . . . . . 9 (𝜑 → ω ∈ (On ∖ 1o))
34 omcl 8573 . . . . . . . . . 10 ((𝑊 ∈ On ∧ 2o ∈ On) → (𝑊 ·o 2o) ∈ On)
3520, 9, 34sylancl 586 . . . . . . . . 9 (𝜑 → (𝑊 ·o 2o) ∈ On)
36 omcl 8573 . . . . . . . . . 10 ((2o ∈ On ∧ 𝑊 ∈ On) → (2o ·o 𝑊) ∈ On)
3712, 20, 36syl2anc 584 . . . . . . . . 9 (𝜑 → (2o ·o 𝑊) ∈ On)
38 fvresi 7193 . . . . . . . . . 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 9736 . . . . . . . 8 (𝜑𝐽:(ω ↑o (𝑊 ·o 2o))–1-1-onto→(ω ↑o (2o ·o 𝑊)))
43 oeoe 8636 . . . . . . . . . 10 ((ω ∈ On ∧ 2o ∈ On ∧ 𝑊 ∈ On) → ((ω ↑o 2o) ↑o 𝑊) = (ω ↑o (2o ·o 𝑊)))
447, 12, 20, 43mp3an2i 1465 . . . . . . . . 9 (𝜑 → ((ω ↑o 2o) ↑o 𝑊) = (ω ↑o (2o ·o 𝑊)))
4544f1oeq3d 6846 . . . . . . . 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))))
4642, 45mpbird 257 . . . . . . 7 (𝜑𝐽:(ω ↑o (𝑊 ·o 2o))–1-1-onto→((ω ↑o 2o) ↑o 𝑊))
47 f1oco 6872 . . . . . . 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))
4824, 46, 47syl2anc 584 . . . . . 6 (𝜑 → (𝐻𝐽):(ω ↑o (𝑊 ·o 2o))–1-1-onto→(ω ↑o 𝑊))
49 df-2o 8506 . . . . . . . . . . . 12 2o = suc 1o
5049oveq2i 7442 . . . . . . . . . . 11 (𝑊 ·o 2o) = (𝑊 ·o suc 1o)
51 1on 8517 . . . . . . . . . . . 12 1o ∈ On
52 omsuc 8563 . . . . . . . . . . . 12 ((𝑊 ∈ On ∧ 1o ∈ On) → (𝑊 ·o suc 1o) = ((𝑊 ·o 1o) +o 𝑊))
5320, 51, 52sylancl 586 . . . . . . . . . . 11 (𝜑 → (𝑊 ·o suc 1o) = ((𝑊 ·o 1o) +o 𝑊))
5450, 53eqtrid 2787 . . . . . . . . . 10 (𝜑 → (𝑊 ·o 2o) = ((𝑊 ·o 1o) +o 𝑊))
55 om1 8579 . . . . . . . . . . . 12 (𝑊 ∈ On → (𝑊 ·o 1o) = 𝑊)
5620, 55syl 17 . . . . . . . . . . 11 (𝜑 → (𝑊 ·o 1o) = 𝑊)
5756oveq1d 7446 . . . . . . . . . 10 (𝜑 → ((𝑊 ·o 1o) +o 𝑊) = (𝑊 +o 𝑊))
5854, 57eqtrd 2775 . . . . . . . . 9 (𝜑 → (𝑊 ·o 2o) = (𝑊 +o 𝑊))
5958oveq2d 7447 . . . . . . . 8 (𝜑 → (ω ↑o (𝑊 ·o 2o)) = (ω ↑o (𝑊 +o 𝑊)))
60 oeoa 8634 . . . . . . . . 9 ((ω ∈ On ∧ 𝑊 ∈ On ∧ 𝑊 ∈ On) → (ω ↑o (𝑊 +o 𝑊)) = ((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
617, 20, 20, 60mp3an2i 1465 . . . . . . . 8 (𝜑 → (ω ↑o (𝑊 +o 𝑊)) = ((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
6259, 61eqtrd 2775 . . . . . . 7 (𝜑 → (ω ↑o (𝑊 ·o 2o)) = ((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
6362f1oeq2d 6845 . . . . . 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)))
6448, 63mpbid 232 . . . . 5 (𝜑 → (𝐻𝐽):((ω ↑o 𝑊) ·o (ω ↑o 𝑊))–1-1-onto→(ω ↑o 𝑊))
65 oecl 8574 . . . . . . 7 ((ω ∈ On ∧ 𝑊 ∈ On) → (ω ↑o 𝑊) ∈ On)
668, 20, 65syl2anc 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))
6867omxpenlem 9112 . . . . . 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)))
6966, 66, 68syl2anc 584 . . . . 5 (𝜑𝑍:((ω ↑o 𝑊) × (ω ↑o 𝑊))–1-1-onto→((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
70 f1oco 6872 . . . . 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))
7164, 69, 70syl2anc 584 . . . 4 (𝜑 → ((𝐻𝐽) ∘ 𝑍):((ω ↑o 𝑊) × (ω ↑o 𝑊))–1-1-onto→(ω ↑o 𝑊))
72 f1of 6849 . . . . . . . . . 10 (𝑁:𝐴1-1-onto→(ω ↑o 𝑊) → 𝑁:𝐴⟶(ω ↑o 𝑊))
731, 72syl 17 . . . . . . . . 9 (𝜑𝑁:𝐴⟶(ω ↑o 𝑊))
7473feqmptd 6977 . . . . . . . 8 (𝜑𝑁 = (𝑥𝐴 ↦ (𝑁𝑥)))
7574f1oeq1d 6844 . . . . . . 7 (𝜑 → (𝑁:𝐴1-1-onto→(ω ↑o 𝑊) ↔ (𝑥𝐴 ↦ (𝑁𝑥)):𝐴1-1-onto→(ω ↑o 𝑊)))
761, 75mpbid 232 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (𝑁𝑥)):𝐴1-1-onto→(ω ↑o 𝑊))
7773feqmptd 6977 . . . . . . . 8 (𝜑𝑁 = (𝑦𝐴 ↦ (𝑁𝑦)))
7877f1oeq1d 6844 . . . . . . 7 (𝜑 → (𝑁:𝐴1-1-onto→(ω ↑o 𝑊) ↔ (𝑦𝐴 ↦ (𝑁𝑦)):𝐴1-1-onto→(ω ↑o 𝑊)))
791, 78mpbid 232 . . . . . 6 (𝜑 → (𝑦𝐴 ↦ (𝑁𝑦)):𝐴1-1-onto→(ω ↑o 𝑊))
8076, 79xpf1o 9178 . . . . 5 (𝜑 → (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩):(𝐴 × 𝐴)–1-1-onto→((ω ↑o 𝑊) × (ω ↑o 𝑊)))
81 infxpenc.t . . . . . 6 𝑇 = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩)
82 f1oeq1 6837 . . . . . 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))))
8381, 82ax-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)))
8480, 83sylibr 234 . . . 4 (𝜑𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑o 𝑊) × (ω ↑o 𝑊)))
85 f1oco 6872 . . . 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))
8671, 84, 85syl2anc 584 . . 3 (𝜑 → (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑o 𝑊))
87 f1oco 6872 . . 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)
883, 86, 87syl2anc 584 . 2 (𝜑 → (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto𝐴)
89 infxpenc.g . . 3 𝐺 = (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇))
90 f1oeq1 6837 . . 3 (𝐺 = (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)) → (𝐺:(𝐴 × 𝐴)–1-1-onto𝐴 ↔ (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto𝐴))
9189, 90ax-mp 5 . 2 (𝐺:(𝐴 × 𝐴)–1-1-onto𝐴 ↔ (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto𝐴)
9288, 91sylibr 234 1 (𝜑𝐺:(𝐴 × 𝐴)–1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  {crab 3433  cdif 3960  wss 3963  c0 4339  cop 4637   class class class wbr 5148  cmpt 5231   I cid 5582   × cxp 5687  ccnv 5688  cres 5691  ccom 5693  Oncon0 6386  suc csuc 6388  wf 6559  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  cmpo 7433  ωcom 7887  1oc1o 8498  2oc2o 8499   +o coa 8502   ·o comu 8503  o coe 8504  m cmap 8865   finSupp cfsupp 9399   CNF ccnf 9699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqom 8487  df-1o 8505  df-2o 8506  df-oadd 8509  df-omul 8510  df-oexp 8511  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-oi 9548  df-cnf 9700
This theorem is referenced by:  infxpenc2lem2  10058
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