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Theorem infxpenc 9432
 Description: A canonical version of infxpen 9428, by a completely different approach (although it uses infxpen 9428 via xpomen 9429). Using Cantor's normal form, we can show that 𝐴 ↑o 𝐵 respects equinumerosity (oef1o 9148), 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 9156.) (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 6603 . . . 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 6628 . . . . . . . . 9 ( I ↾ 𝑊):𝑊1-1-onto𝑊
65a1i 11 . . . . . . . 8 (𝜑 → ( I ↾ 𝑊):𝑊1-1-onto𝑊)
7 omelon 9096 . . . . . . . . . . 11 ω ∈ On
87a1i 11 . . . . . . . . . 10 (𝜑 → ω ∈ On)
9 2on 8097 . . . . . . . . . 10 2o ∈ On
10 oecl 8148 . . . . . . . . . 10 ((ω ∈ On ∧ 2o ∈ On) → (ω ↑o 2o) ∈ On)
118, 9, 10sylancl 589 . . . . . . . . 9 (𝜑 → (ω ↑o 2o) ∈ On)
129a1i 11 . . . . . . . . . 10 (𝜑 → 2o ∈ On)
13 peano1 7584 . . . . . . . . . . 11 ∅ ∈ ω
1413a1i 11 . . . . . . . . . 10 (𝜑 → ∅ ∈ ω)
15 oen0 8198 . . . . . . . . . 10 (((ω ∈ On ∧ 2o ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o 2o))
168, 12, 14, 15syl21anc 836 . . . . . . . . 9 (𝜑 → ∅ ∈ (ω ↑o 2o))
17 ondif1 8112 . . . . . . . . 9 ((ω ↑o 2o) ∈ (On ∖ 1o) ↔ ((ω ↑o 2o) ∈ On ∧ ∅ ∈ (ω ↑o 2o)))
1811, 16, 17sylanbrc 586 . . . . . . . 8 (𝜑 → (ω ↑o 2o) ∈ (On ∖ 1o))
19 infxpenc.3 . . . . . . . . 9 (𝜑𝑊 ∈ (On ∖ 1o))
2019eldifad 3893 . . . . . . . 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 9148 . . . . . . 7 (𝜑𝐻:((ω ↑o 2o) ↑o 𝑊)–1-1-onto→(ω ↑o 𝑊))
25 f1oi 6628 . . . . . . . . . 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 8606 . . . . . . . . . 10 ((𝑊 ∈ On ∧ 2o ∈ On) → (𝑌𝑋):(𝑊 ·o 2o)–1-1-onto→(2o ·o 𝑊))
3020, 9, 29sylancl 589 . . . . . . . . 9 (𝜑 → (𝑌𝑋):(𝑊 ·o 2o)–1-1-onto→(2o ·o 𝑊))
31 ondif1 8112 . . . . . . . . . . 11 (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
327, 13, 31mpbir2an 710 . . . . . . . . . 10 ω ∈ (On ∖ 1o)
3332a1i 11 . . . . . . . . 9 (𝜑 → ω ∈ (On ∖ 1o))
34 omcl 8147 . . . . . . . . . 10 ((𝑊 ∈ On ∧ 2o ∈ On) → (𝑊 ·o 2o) ∈ On)
3520, 9, 34sylancl 589 . . . . . . . . 9 (𝜑 → (𝑊 ·o 2o) ∈ On)
36 omcl 8147 . . . . . . . . . 10 ((2o ∈ On ∧ 𝑊 ∈ On) → (2o ·o 𝑊) ∈ On)
3712, 20, 36syl2anc 587 . . . . . . . . 9 (𝜑 → (2o ·o 𝑊) ∈ On)
38 fvresi 6913 . . . . . . . . . 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 9148 . . . . . . . 8 (𝜑𝐽:(ω ↑o (𝑊 ·o 2o))–1-1-onto→(ω ↑o (2o ·o 𝑊)))
43 oeoe 8211 . . . . . . . . . 10 ((ω ∈ On ∧ 2o ∈ On ∧ 𝑊 ∈ On) → ((ω ↑o 2o) ↑o 𝑊) = (ω ↑o (2o ·o 𝑊)))
447, 12, 20, 43mp3an2i 1463 . . . . . . . . 9 (𝜑 → ((ω ↑o 2o) ↑o 𝑊) = (ω ↑o (2o ·o 𝑊)))
4544f1oeq3d 6588 . . . . . . . 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 260 . . . . . . 7 (𝜑𝐽:(ω ↑o (𝑊 ·o 2o))–1-1-onto→((ω ↑o 2o) ↑o 𝑊))
47 f1oco 6613 . . . . . . 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 587 . . . . . 6 (𝜑 → (𝐻𝐽):(ω ↑o (𝑊 ·o 2o))–1-1-onto→(ω ↑o 𝑊))
49 df-2o 8089 . . . . . . . . . . . 12 2o = suc 1o
5049oveq2i 7147 . . . . . . . . . . 11 (𝑊 ·o 2o) = (𝑊 ·o suc 1o)
51 1on 8095 . . . . . . . . . . . 12 1o ∈ On
52 omsuc 8137 . . . . . . . . . . . 12 ((𝑊 ∈ On ∧ 1o ∈ On) → (𝑊 ·o suc 1o) = ((𝑊 ·o 1o) +o 𝑊))
5320, 51, 52sylancl 589 . . . . . . . . . . 11 (𝜑 → (𝑊 ·o suc 1o) = ((𝑊 ·o 1o) +o 𝑊))
5450, 53syl5eq 2845 . . . . . . . . . 10 (𝜑 → (𝑊 ·o 2o) = ((𝑊 ·o 1o) +o 𝑊))
55 om1 8154 . . . . . . . . . . . 12 (𝑊 ∈ On → (𝑊 ·o 1o) = 𝑊)
5620, 55syl 17 . . . . . . . . . . 11 (𝜑 → (𝑊 ·o 1o) = 𝑊)
5756oveq1d 7151 . . . . . . . . . 10 (𝜑 → ((𝑊 ·o 1o) +o 𝑊) = (𝑊 +o 𝑊))
5854, 57eqtrd 2833 . . . . . . . . 9 (𝜑 → (𝑊 ·o 2o) = (𝑊 +o 𝑊))
5958oveq2d 7152 . . . . . . . 8 (𝜑 → (ω ↑o (𝑊 ·o 2o)) = (ω ↑o (𝑊 +o 𝑊)))
60 oeoa 8209 . . . . . . . . 9 ((ω ∈ On ∧ 𝑊 ∈ On ∧ 𝑊 ∈ On) → (ω ↑o (𝑊 +o 𝑊)) = ((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
617, 20, 20, 60mp3an2i 1463 . . . . . . . 8 (𝜑 → (ω ↑o (𝑊 +o 𝑊)) = ((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
6259, 61eqtrd 2833 . . . . . . 7 (𝜑 → (ω ↑o (𝑊 ·o 2o)) = ((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
6362f1oeq2d 6587 . . . . . 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 235 . . . . 5 (𝜑 → (𝐻𝐽):((ω ↑o 𝑊) ·o (ω ↑o 𝑊))–1-1-onto→(ω ↑o 𝑊))
65 oecl 8148 . . . . . . 7 ((ω ∈ On ∧ 𝑊 ∈ On) → (ω ↑o 𝑊) ∈ On)
668, 20, 65syl2anc 587 . . . . . 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 8604 . . . . . 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 587 . . . . 5 (𝜑𝑍:((ω ↑o 𝑊) × (ω ↑o 𝑊))–1-1-onto→((ω ↑o 𝑊) ·o (ω ↑o 𝑊)))
70 f1oco 6613 . . . . 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 587 . . . 4 (𝜑 → ((𝐻𝐽) ∘ 𝑍):((ω ↑o 𝑊) × (ω ↑o 𝑊))–1-1-onto→(ω ↑o 𝑊))
72 f1of 6591 . . . . . . . . . 10 (𝑁:𝐴1-1-onto→(ω ↑o 𝑊) → 𝑁:𝐴⟶(ω ↑o 𝑊))
731, 72syl 17 . . . . . . . . 9 (𝜑𝑁:𝐴⟶(ω ↑o 𝑊))
7473feqmptd 6709 . . . . . . . 8 (𝜑𝑁 = (𝑥𝐴 ↦ (𝑁𝑥)))
75 f1oeq1 6580 . . . . . . . 8 (𝑁 = (𝑥𝐴 ↦ (𝑁𝑥)) → (𝑁:𝐴1-1-onto→(ω ↑o 𝑊) ↔ (𝑥𝐴 ↦ (𝑁𝑥)):𝐴1-1-onto→(ω ↑o 𝑊)))
7674, 75syl 17 . . . . . . 7 (𝜑 → (𝑁:𝐴1-1-onto→(ω ↑o 𝑊) ↔ (𝑥𝐴 ↦ (𝑁𝑥)):𝐴1-1-onto→(ω ↑o 𝑊)))
771, 76mpbid 235 . . . . . 6 (𝜑 → (𝑥𝐴 ↦ (𝑁𝑥)):𝐴1-1-onto→(ω ↑o 𝑊))
7873feqmptd 6709 . . . . . . . 8 (𝜑𝑁 = (𝑦𝐴 ↦ (𝑁𝑦)))
79 f1oeq1 6580 . . . . . . . 8 (𝑁 = (𝑦𝐴 ↦ (𝑁𝑦)) → (𝑁:𝐴1-1-onto→(ω ↑o 𝑊) ↔ (𝑦𝐴 ↦ (𝑁𝑦)):𝐴1-1-onto→(ω ↑o 𝑊)))
8078, 79syl 17 . . . . . . 7 (𝜑 → (𝑁:𝐴1-1-onto→(ω ↑o 𝑊) ↔ (𝑦𝐴 ↦ (𝑁𝑦)):𝐴1-1-onto→(ω ↑o 𝑊)))
811, 80mpbid 235 . . . . . 6 (𝜑 → (𝑦𝐴 ↦ (𝑁𝑦)):𝐴1-1-onto→(ω ↑o 𝑊))
8277, 81xpf1o 8666 . . . . 5 (𝜑 → (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩):(𝐴 × 𝐴)–1-1-onto→((ω ↑o 𝑊) × (ω ↑o 𝑊)))
83 infxpenc.t . . . . . 6 𝑇 = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝑁𝑥), (𝑁𝑦)⟩)
84 f1oeq1 6580 . . . . . 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))))
8583, 84ax-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)))
8682, 85sylibr 237 . . . 4 (𝜑𝑇:(𝐴 × 𝐴)–1-1-onto→((ω ↑o 𝑊) × (ω ↑o 𝑊)))
87 f1oco 6613 . . . 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))
8871, 86, 87syl2anc 587 . . 3 (𝜑 → (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇):(𝐴 × 𝐴)–1-1-onto→(ω ↑o 𝑊))
89 f1oco 6613 . . 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)
903, 88, 89syl2anc 587 . 2 (𝜑 → (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto𝐴)
91 infxpenc.g . . 3 𝐺 = (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇))
92 f1oeq1 6580 . . 3 (𝐺 = (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)) → (𝐺:(𝐴 × 𝐴)–1-1-onto𝐴 ↔ (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto𝐴))
9391, 92ax-mp 5 . 2 (𝐺:(𝐴 × 𝐴)–1-1-onto𝐴 ↔ (𝑁 ∘ (((𝐻𝐽) ∘ 𝑍) ∘ 𝑇)):(𝐴 × 𝐴)–1-1-onto𝐴)
9490, 93sylibr 237 1 (𝜑𝐺:(𝐴 × 𝐴)–1-1-onto𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2111  {crab 3110   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  ⟨cop 4531   class class class wbr 5031   ↦ cmpt 5111   I cid 5425   × cxp 5518  ◡ccnv 5519   ↾ cres 5522   ∘ ccom 5524  Oncon0 6160  suc csuc 6162  ⟶wf 6321  –1-1-onto→wf1o 6324  ‘cfv 6325  (class class class)co 7136   ∈ cmpo 7138  ωcom 7563  1oc1o 8081  2oc2o 8082   +o coa 8085   ·o comu 8086   ↑o coe 8087   ↑m cmap 8392   finSupp cfsupp 8820   CNF ccnf 9111 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-inf2 9091 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-se 5480  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-isom 6334  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-supp 7817  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-seqom 8070  df-1o 8088  df-2o 8089  df-oadd 8092  df-omul 8093  df-oexp 8094  df-er 8275  df-map 8394  df-en 8496  df-dom 8497  df-sdom 8498  df-fin 8499  df-fsupp 8821  df-oi 8961  df-cnf 9112 This theorem is referenced by:  infxpenc2lem2  9434
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