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Theorem cnfcom 8760
 Description: Any ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. Here we show that bijection explicitly. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
Hypotheses
Ref Expression
cnfcom.s 𝑆 = dom (ω CNF 𝐴)
cnfcom.a (𝜑𝐴 ∈ On)
cnfcom.b (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))
cnfcom.f 𝐹 = ((ω CNF 𝐴)‘𝐵)
cnfcom.g 𝐺 = OrdIso( E , (𝐹 supp ∅))
cnfcom.h 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)
cnfcom.t 𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
cnfcom.m 𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))
cnfcom.k 𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))
cnfcom.1 (𝜑𝐼 ∈ dom 𝐺)
Assertion
Ref Expression
cnfcom (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
Distinct variable groups:   𝑥,𝑘,𝑧,𝐴   𝑘,𝐼,𝑥,𝑧   𝑥,𝑀   𝑓,𝑘,𝑥,𝑧,𝐹   𝑧,𝑇   𝑓,𝐺,𝑘,𝑥,𝑧   𝑓,𝐻,𝑥   𝑆,𝑘,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑧,𝑓,𝑘)   𝐴(𝑓)   𝐵(𝑥,𝑧,𝑓,𝑘)   𝑆(𝑥,𝑓)   𝑇(𝑥,𝑓,𝑘)   𝐻(𝑧,𝑘)   𝐼(𝑓)   𝐾(𝑥,𝑧,𝑓,𝑘)   𝑀(𝑧,𝑓,𝑘)

Proof of Theorem cnfcom
Dummy variables 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnfcom.1 . 2 (𝜑𝐼 ∈ dom 𝐺)
2 cnfcom.s . . . . . 6 𝑆 = dom (ω CNF 𝐴)
3 omelon 8706 . . . . . . 7 ω ∈ On
43a1i 11 . . . . . 6 (𝜑 → ω ∈ On)
5 cnfcom.a . . . . . 6 (𝜑𝐴 ∈ On)
6 cnfcom.g . . . . . 6 𝐺 = OrdIso( E , (𝐹 supp ∅))
7 cnfcom.f . . . . . . 7 𝐹 = ((ω CNF 𝐴)‘𝐵)
82, 4, 5cantnff1o 8756 . . . . . . . . 9 (𝜑 → (ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴))
9 f1ocnv 6290 . . . . . . . . 9 ((ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴) → (ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆)
10 f1of 6278 . . . . . . . . 9 ((ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆)
118, 9, 103syl 18 . . . . . . . 8 (𝜑(ω CNF 𝐴):(ω ↑𝑜 𝐴)⟶𝑆)
12 cnfcom.b . . . . . . . 8 (𝜑𝐵 ∈ (ω ↑𝑜 𝐴))
1311, 12ffvelrnd 6503 . . . . . . 7 (𝜑 → ((ω CNF 𝐴)‘𝐵) ∈ 𝑆)
147, 13syl5eqel 2853 . . . . . 6 (𝜑𝐹𝑆)
152, 4, 5, 6, 14cantnfcl 8727 . . . . 5 (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
1615simprd 477 . . . 4 (𝜑 → dom 𝐺 ∈ ω)
17 elnn 7221 . . . 4 ((𝐼 ∈ dom 𝐺 ∧ dom 𝐺 ∈ ω) → 𝐼 ∈ ω)
181, 16, 17syl2anc 565 . . 3 (𝜑𝐼 ∈ ω)
19 eleq1 2837 . . . . . 6 (𝑤 = 𝐼 → (𝑤 ∈ dom 𝐺𝐼 ∈ dom 𝐺))
20 suceq 5933 . . . . . . . 8 (𝑤 = 𝐼 → suc 𝑤 = suc 𝐼)
2120fveq2d 6336 . . . . . . 7 (𝑤 = 𝐼 → (𝑇‘suc 𝑤) = (𝑇‘suc 𝐼))
2220fveq2d 6336 . . . . . . 7 (𝑤 = 𝐼 → (𝐻‘suc 𝑤) = (𝐻‘suc 𝐼))
23 fveq2 6332 . . . . . . . . 9 (𝑤 = 𝐼 → (𝐺𝑤) = (𝐺𝐼))
2423oveq2d 6808 . . . . . . . 8 (𝑤 = 𝐼 → (ω ↑𝑜 (𝐺𝑤)) = (ω ↑𝑜 (𝐺𝐼)))
2523fveq2d 6336 . . . . . . . 8 (𝑤 = 𝐼 → (𝐹‘(𝐺𝑤)) = (𝐹‘(𝐺𝐼)))
2624, 25oveq12d 6810 . . . . . . 7 (𝑤 = 𝐼 → ((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤))) = ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
2721, 22, 26f1oeq123d 6274 . . . . . 6 (𝑤 = 𝐼 → ((𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤))) ↔ (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))))
2819, 27imbi12d 333 . . . . 5 (𝑤 = 𝐼 → ((𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤)))) ↔ (𝐼 ∈ dom 𝐺 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))))
2928imbi2d 329 . . . 4 (𝑤 = 𝐼 → ((𝜑 → (𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤))))) ↔ (𝜑 → (𝐼 ∈ dom 𝐺 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))))))
30 eleq1 2837 . . . . . 6 (𝑤 = ∅ → (𝑤 ∈ dom 𝐺 ↔ ∅ ∈ dom 𝐺))
31 suceq 5933 . . . . . . . 8 (𝑤 = ∅ → suc 𝑤 = suc ∅)
3231fveq2d 6336 . . . . . . 7 (𝑤 = ∅ → (𝑇‘suc 𝑤) = (𝑇‘suc ∅))
3331fveq2d 6336 . . . . . . 7 (𝑤 = ∅ → (𝐻‘suc 𝑤) = (𝐻‘suc ∅))
34 fveq2 6332 . . . . . . . . 9 (𝑤 = ∅ → (𝐺𝑤) = (𝐺‘∅))
3534oveq2d 6808 . . . . . . . 8 (𝑤 = ∅ → (ω ↑𝑜 (𝐺𝑤)) = (ω ↑𝑜 (𝐺‘∅)))
3634fveq2d 6336 . . . . . . . 8 (𝑤 = ∅ → (𝐹‘(𝐺𝑤)) = (𝐹‘(𝐺‘∅)))
3735, 36oveq12d 6810 . . . . . . 7 (𝑤 = ∅ → ((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤))) = ((ω ↑𝑜 (𝐺‘∅)) ·𝑜 (𝐹‘(𝐺‘∅))))
3832, 33, 37f1oeq123d 6274 . . . . . 6 (𝑤 = ∅ → ((𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤))) ↔ (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑𝑜 (𝐺‘∅)) ·𝑜 (𝐹‘(𝐺‘∅)))))
3930, 38imbi12d 333 . . . . 5 (𝑤 = ∅ → ((𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤)))) ↔ (∅ ∈ dom 𝐺 → (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑𝑜 (𝐺‘∅)) ·𝑜 (𝐹‘(𝐺‘∅))))))
40 eleq1 2837 . . . . . 6 (𝑤 = 𝑦 → (𝑤 ∈ dom 𝐺𝑦 ∈ dom 𝐺))
41 suceq 5933 . . . . . . . 8 (𝑤 = 𝑦 → suc 𝑤 = suc 𝑦)
4241fveq2d 6336 . . . . . . 7 (𝑤 = 𝑦 → (𝑇‘suc 𝑤) = (𝑇‘suc 𝑦))
4341fveq2d 6336 . . . . . . 7 (𝑤 = 𝑦 → (𝐻‘suc 𝑤) = (𝐻‘suc 𝑦))
44 fveq2 6332 . . . . . . . . 9 (𝑤 = 𝑦 → (𝐺𝑤) = (𝐺𝑦))
4544oveq2d 6808 . . . . . . . 8 (𝑤 = 𝑦 → (ω ↑𝑜 (𝐺𝑤)) = (ω ↑𝑜 (𝐺𝑦)))
4644fveq2d 6336 . . . . . . . 8 (𝑤 = 𝑦 → (𝐹‘(𝐺𝑤)) = (𝐹‘(𝐺𝑦)))
4745, 46oveq12d 6810 . . . . . . 7 (𝑤 = 𝑦 → ((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤))) = ((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))
4842, 43, 47f1oeq123d 6274 . . . . . 6 (𝑤 = 𝑦 → ((𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤))) ↔ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦)))))
4940, 48imbi12d 333 . . . . 5 (𝑤 = 𝑦 → ((𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤)))) ↔ (𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))))
50 eleq1 2837 . . . . . 6 (𝑤 = suc 𝑦 → (𝑤 ∈ dom 𝐺 ↔ suc 𝑦 ∈ dom 𝐺))
51 suceq 5933 . . . . . . . 8 (𝑤 = suc 𝑦 → suc 𝑤 = suc suc 𝑦)
5251fveq2d 6336 . . . . . . 7 (𝑤 = suc 𝑦 → (𝑇‘suc 𝑤) = (𝑇‘suc suc 𝑦))
5351fveq2d 6336 . . . . . . 7 (𝑤 = suc 𝑦 → (𝐻‘suc 𝑤) = (𝐻‘suc suc 𝑦))
54 fveq2 6332 . . . . . . . . 9 (𝑤 = suc 𝑦 → (𝐺𝑤) = (𝐺‘suc 𝑦))
5554oveq2d 6808 . . . . . . . 8 (𝑤 = suc 𝑦 → (ω ↑𝑜 (𝐺𝑤)) = (ω ↑𝑜 (𝐺‘suc 𝑦)))
5654fveq2d 6336 . . . . . . . 8 (𝑤 = suc 𝑦 → (𝐹‘(𝐺𝑤)) = (𝐹‘(𝐺‘suc 𝑦)))
5755, 56oveq12d 6810 . . . . . . 7 (𝑤 = suc 𝑦 → ((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤))) = ((ω ↑𝑜 (𝐺‘suc 𝑦)) ·𝑜 (𝐹‘(𝐺‘suc 𝑦))))
5852, 53, 57f1oeq123d 6274 . . . . . 6 (𝑤 = suc 𝑦 → ((𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤))) ↔ (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺‘suc 𝑦)) ·𝑜 (𝐹‘(𝐺‘suc 𝑦)))))
5950, 58imbi12d 333 . . . . 5 (𝑤 = suc 𝑦 → ((𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤)))) ↔ (suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺‘suc 𝑦)) ·𝑜 (𝐹‘(𝐺‘suc 𝑦))))))
605adantr 466 . . . . . . 7 ((𝜑 ∧ ∅ ∈ dom 𝐺) → 𝐴 ∈ On)
6112adantr 466 . . . . . . 7 ((𝜑 ∧ ∅ ∈ dom 𝐺) → 𝐵 ∈ (ω ↑𝑜 𝐴))
62 cnfcom.h . . . . . . 7 𝐻 = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +𝑜 𝑧)), ∅)
63 cnfcom.t . . . . . . 7 𝑇 = seq𝜔((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)
64 cnfcom.m . . . . . . 7 𝑀 = ((ω ↑𝑜 (𝐺𝑘)) ·𝑜 (𝐹‘(𝐺𝑘)))
65 cnfcom.k . . . . . . 7 𝐾 = ((𝑥𝑀 ↦ (dom 𝑓 +𝑜 𝑥)) ∪ (𝑥 ∈ dom 𝑓 ↦ (𝑀 +𝑜 𝑥)))
66 simpr 471 . . . . . . 7 ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ dom 𝐺)
673a1i 11 . . . . . . . 8 ((𝜑 ∧ ∅ ∈ dom 𝐺) → ω ∈ On)
68 suppssdm 7458 . . . . . . . . . . 11 (𝐹 supp ∅) ⊆ dom 𝐹
692, 4, 5cantnfs 8726 . . . . . . . . . . . . . 14 (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
7014, 69mpbid 222 . . . . . . . . . . . . 13 (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
7170simpld 476 . . . . . . . . . . . 12 (𝜑𝐹:𝐴⟶ω)
72 fdm 6191 . . . . . . . . . . . 12 (𝐹:𝐴⟶ω → dom 𝐹 = 𝐴)
7371, 72syl 17 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = 𝐴)
7468, 73syl5sseq 3800 . . . . . . . . . 10 (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
75 onss 7136 . . . . . . . . . . 11 (𝐴 ∈ On → 𝐴 ⊆ On)
765, 75syl 17 . . . . . . . . . 10 (𝜑𝐴 ⊆ On)
7774, 76sstrd 3760 . . . . . . . . 9 (𝜑 → (𝐹 supp ∅) ⊆ On)
786oif 8590 . . . . . . . . . 10 𝐺:dom 𝐺⟶(𝐹 supp ∅)
7978ffvelrni 6501 . . . . . . . . 9 (∅ ∈ dom 𝐺 → (𝐺‘∅) ∈ (𝐹 supp ∅))
80 ssel2 3745 . . . . . . . . 9 (((𝐹 supp ∅) ⊆ On ∧ (𝐺‘∅) ∈ (𝐹 supp ∅)) → (𝐺‘∅) ∈ On)
8177, 79, 80syl2an 575 . . . . . . . 8 ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝐺‘∅) ∈ On)
82 peano1 7231 . . . . . . . . 9 ∅ ∈ ω
8382a1i 11 . . . . . . . 8 ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ ω)
84 oen0 7819 . . . . . . . 8 (((ω ∈ On ∧ (𝐺‘∅) ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑𝑜 (𝐺‘∅)))
8567, 81, 83, 84syl21anc 1474 . . . . . . 7 ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ (ω ↑𝑜 (𝐺‘∅)))
86 0ex 4921 . . . . . . . . 9 ∅ ∈ V
8763seqom0g 7703 . . . . . . . . 9 (∅ ∈ V → (𝑇‘∅) = ∅)
8886, 87ax-mp 5 . . . . . . . 8 (𝑇‘∅) = ∅
89 f1o0 6314 . . . . . . . . . 10 ∅:∅–1-1-onto→∅
9062seqom0g 7703 . . . . . . . . . . 11 (∅ ∈ V → (𝐻‘∅) = ∅)
91 f1oeq2 6269 . . . . . . . . . . 11 ((𝐻‘∅) = ∅ → (∅:(𝐻‘∅)–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
9286, 90, 91mp2b 10 . . . . . . . . . 10 (∅:(𝐻‘∅)–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅)
9389, 92mpbir 221 . . . . . . . . 9 ∅:(𝐻‘∅)–1-1-onto→∅
94 f1oeq1 6268 . . . . . . . . 9 ((𝑇‘∅) = ∅ → ((𝑇‘∅):(𝐻‘∅)–1-1-onto→∅ ↔ ∅:(𝐻‘∅)–1-1-onto→∅))
9593, 94mpbiri 248 . . . . . . . 8 ((𝑇‘∅) = ∅ → (𝑇‘∅):(𝐻‘∅)–1-1-onto→∅)
9688, 95mp1i 13 . . . . . . 7 ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝑇‘∅):(𝐻‘∅)–1-1-onto→∅)
972, 60, 61, 7, 6, 62, 63, 64, 65, 66, 85, 96cnfcomlem 8759 . . . . . 6 ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑𝑜 (𝐺‘∅)) ·𝑜 (𝐹‘(𝐺‘∅))))
9897ex 397 . . . . 5 (𝜑 → (∅ ∈ dom 𝐺 → (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑𝑜 (𝐺‘∅)) ·𝑜 (𝐹‘(𝐺‘∅)))))
996oicl 8589 . . . . . . . . . 10 Ord dom 𝐺
100 ordtr 5880 . . . . . . . . . 10 (Ord dom 𝐺 → Tr dom 𝐺)
10199, 100ax-mp 5 . . . . . . . . 9 Tr dom 𝐺
102 trsuc 5953 . . . . . . . . 9 ((Tr dom 𝐺 ∧ suc 𝑦 ∈ dom 𝐺) → 𝑦 ∈ dom 𝐺)
103101, 102mpan 662 . . . . . . . 8 (suc 𝑦 ∈ dom 𝐺𝑦 ∈ dom 𝐺)
104103imim1i 63 . . . . . . 7 ((𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦)))) → (suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦)))))
1055ad2antrr 697 . . . . . . . . . 10 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → 𝐴 ∈ On)
10612ad2antrr 697 . . . . . . . . . 10 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → 𝐵 ∈ (ω ↑𝑜 𝐴))
107 simprl 746 . . . . . . . . . 10 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → suc 𝑦 ∈ dom 𝐺)
10876ad2antrr 697 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → 𝐴 ⊆ On)
10974ad2antrr 697 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (𝐹 supp ∅) ⊆ 𝐴)
11078ffvelrni 6501 . . . . . . . . . . . . . . . . 17 (suc 𝑦 ∈ dom 𝐺 → (𝐺‘suc 𝑦) ∈ (𝐹 supp ∅))
111110ad2antrl 699 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (𝐺‘suc 𝑦) ∈ (𝐹 supp ∅))
112109, 111sseldd 3751 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (𝐺‘suc 𝑦) ∈ 𝐴)
113108, 112sseldd 3751 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (𝐺‘suc 𝑦) ∈ On)
114 eloni 5876 . . . . . . . . . . . . . 14 ((𝐺‘suc 𝑦) ∈ On → Ord (𝐺‘suc 𝑦))
115113, 114syl 17 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → Ord (𝐺‘suc 𝑦))
116 vex 3352 . . . . . . . . . . . . . . 15 𝑦 ∈ V
117116sucid 5947 . . . . . . . . . . . . . 14 𝑦 ∈ suc 𝑦
1185, 74ssexd 4936 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹 supp ∅) ∈ V)
11915simpld 476 . . . . . . . . . . . . . . . . . 18 (𝜑 → E We (𝐹 supp ∅))
1206oiiso 8597 . . . . . . . . . . . . . . . . . 18 (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
121118, 119, 120syl2anc 565 . . . . . . . . . . . . . . . . 17 (𝜑𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
122121ad2antrr 697 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
123103ad2antrl 699 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → 𝑦 ∈ dom 𝐺)
124 isorel 6718 . . . . . . . . . . . . . . . 16 ((𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)) ∧ (𝑦 ∈ dom 𝐺 ∧ suc 𝑦 ∈ dom 𝐺)) → (𝑦 E suc 𝑦 ↔ (𝐺𝑦) E (𝐺‘suc 𝑦)))
125122, 123, 107, 124syl12anc 1473 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (𝑦 E suc 𝑦 ↔ (𝐺𝑦) E (𝐺‘suc 𝑦)))
126116sucex 7157 . . . . . . . . . . . . . . . 16 suc 𝑦 ∈ V
127126epelc 5164 . . . . . . . . . . . . . . 15 (𝑦 E suc 𝑦𝑦 ∈ suc 𝑦)
128 fvex 6342 . . . . . . . . . . . . . . . 16 (𝐺‘suc 𝑦) ∈ V
129128epelc 5164 . . . . . . . . . . . . . . 15 ((𝐺𝑦) E (𝐺‘suc 𝑦) ↔ (𝐺𝑦) ∈ (𝐺‘suc 𝑦))
130125, 127, 1293bitr3g 302 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (𝑦 ∈ suc 𝑦 ↔ (𝐺𝑦) ∈ (𝐺‘suc 𝑦)))
131117, 130mpbii 223 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (𝐺𝑦) ∈ (𝐺‘suc 𝑦))
132 ordsucss 7164 . . . . . . . . . . . . 13 (Ord (𝐺‘suc 𝑦) → ((𝐺𝑦) ∈ (𝐺‘suc 𝑦) → suc (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)))
133115, 131, 132sylc 65 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → suc (𝐺𝑦) ⊆ (𝐺‘suc 𝑦))
13478ffvelrni 6501 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ dom 𝐺 → (𝐺𝑦) ∈ (𝐹 supp ∅))
135123, 134syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (𝐺𝑦) ∈ (𝐹 supp ∅))
136109, 135sseldd 3751 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (𝐺𝑦) ∈ 𝐴)
137108, 136sseldd 3751 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (𝐺𝑦) ∈ On)
138 suceloni 7159 . . . . . . . . . . . . . 14 ((𝐺𝑦) ∈ On → suc (𝐺𝑦) ∈ On)
139137, 138syl 17 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → suc (𝐺𝑦) ∈ On)
1403a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → ω ∈ On)
14182a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → ∅ ∈ ω)
142 oewordi 7824 . . . . . . . . . . . . 13 (((suc (𝐺𝑦) ∈ On ∧ (𝐺‘suc 𝑦) ∈ On ∧ ω ∈ On) ∧ ∅ ∈ ω) → (suc (𝐺𝑦) ⊆ (𝐺‘suc 𝑦) → (ω ↑𝑜 suc (𝐺𝑦)) ⊆ (ω ↑𝑜 (𝐺‘suc 𝑦))))
143139, 113, 140, 141, 142syl31anc 1478 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (suc (𝐺𝑦) ⊆ (𝐺‘suc 𝑦) → (ω ↑𝑜 suc (𝐺𝑦)) ⊆ (ω ↑𝑜 (𝐺‘suc 𝑦))))
144133, 143mpd 15 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (ω ↑𝑜 suc (𝐺𝑦)) ⊆ (ω ↑𝑜 (𝐺‘suc 𝑦)))
14571ad2antrr 697 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → 𝐹:𝐴⟶ω)
146145, 136ffvelrnd 6503 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (𝐹‘(𝐺𝑦)) ∈ ω)
147 nnon 7217 . . . . . . . . . . . . . . 15 ((𝐹‘(𝐺𝑦)) ∈ ω → (𝐹‘(𝐺𝑦)) ∈ On)
148146, 147syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (𝐹‘(𝐺𝑦)) ∈ On)
149 oecl 7770 . . . . . . . . . . . . . . 15 ((ω ∈ On ∧ (𝐺𝑦) ∈ On) → (ω ↑𝑜 (𝐺𝑦)) ∈ On)
150140, 137, 149syl2anc 565 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (ω ↑𝑜 (𝐺𝑦)) ∈ On)
151 oen0 7819 . . . . . . . . . . . . . . 15 (((ω ∈ On ∧ (𝐺𝑦) ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑𝑜 (𝐺𝑦)))
152140, 137, 141, 151syl21anc 1474 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → ∅ ∈ (ω ↑𝑜 (𝐺𝑦)))
153 omord2 7800 . . . . . . . . . . . . . 14 ((((𝐹‘(𝐺𝑦)) ∈ On ∧ ω ∈ On ∧ (ω ↑𝑜 (𝐺𝑦)) ∈ On) ∧ ∅ ∈ (ω ↑𝑜 (𝐺𝑦))) → ((𝐹‘(𝐺𝑦)) ∈ ω ↔ ((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))) ∈ ((ω ↑𝑜 (𝐺𝑦)) ·𝑜 ω)))
154148, 140, 150, 152, 153syl31anc 1478 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → ((𝐹‘(𝐺𝑦)) ∈ ω ↔ ((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))) ∈ ((ω ↑𝑜 (𝐺𝑦)) ·𝑜 ω)))
155146, 154mpbid 222 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → ((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))) ∈ ((ω ↑𝑜 (𝐺𝑦)) ·𝑜 ω))
156 oesuc 7760 . . . . . . . . . . . . 13 ((ω ∈ On ∧ (𝐺𝑦) ∈ On) → (ω ↑𝑜 suc (𝐺𝑦)) = ((ω ↑𝑜 (𝐺𝑦)) ·𝑜 ω))
157140, 137, 156syl2anc 565 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (ω ↑𝑜 suc (𝐺𝑦)) = ((ω ↑𝑜 (𝐺𝑦)) ·𝑜 ω))
158155, 157eleqtrrd 2852 . . . . . . . . . . 11 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → ((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))) ∈ (ω ↑𝑜 suc (𝐺𝑦)))
159144, 158sseldd 3751 . . . . . . . . . 10 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → ((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))) ∈ (ω ↑𝑜 (𝐺‘suc 𝑦)))
160 simprr 748 . . . . . . . . . 10 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))
1612, 105, 106, 7, 6, 62, 63, 64, 65, 107, 159, 160cnfcomlem 8759 . . . . . . . . 9 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))) → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺‘suc 𝑦)) ·𝑜 (𝐹‘(𝐺‘suc 𝑦))))
162161exp32 407 . . . . . . . 8 ((𝜑𝑦 ∈ ω) → (suc 𝑦 ∈ dom 𝐺 → ((𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))) → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺‘suc 𝑦)) ·𝑜 (𝐹‘(𝐺‘suc 𝑦))))))
163162a2d 29 . . . . . . 7 ((𝜑𝑦 ∈ ω) → ((suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦)))) → (suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺‘suc 𝑦)) ·𝑜 (𝐹‘(𝐺‘suc 𝑦))))))
164104, 163syl5 34 . . . . . 6 ((𝜑𝑦 ∈ ω) → ((𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦)))) → (suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺‘suc 𝑦)) ·𝑜 (𝐹‘(𝐺‘suc 𝑦))))))
165164expcom 398 . . . . 5 (𝑦 ∈ ω → (𝜑 → ((𝑦 ∈ dom 𝐺 → (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦)))) → (suc 𝑦 ∈ dom 𝐺 → (𝑇‘suc suc 𝑦):(𝐻‘suc suc 𝑦)–1-1-onto→((ω ↑𝑜 (𝐺‘suc 𝑦)) ·𝑜 (𝐹‘(𝐺‘suc 𝑦)))))))
16639, 49, 59, 98, 165finds2 7240 . . . 4 (𝑤 ∈ ω → (𝜑 → (𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤))))))
16729, 166vtoclga 3421 . . 3 (𝐼 ∈ ω → (𝜑 → (𝐼 ∈ dom 𝐺 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))))
16818, 167mpcom 38 . 2 (𝜑 → (𝐼 ∈ dom 𝐺 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼)))))
1691, 168mpd 15 1 (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1630   ∈ wcel 2144  Vcvv 3349   ∪ cun 3719   ⊆ wss 3721  ∅c0 4061   class class class wbr 4784   ↦ cmpt 4861  Tr wtr 4884   E cep 5161   We wwe 5207  ◡ccnv 5248  dom cdm 5249  Ord word 5865  Oncon0 5866  suc csuc 5868  ⟶wf 6027  –1-1-onto→wf1o 6030  ‘cfv 6031   Isom wiso 6032  (class class class)co 6792   ↦ cmpt2 6794  ωcom 7211   supp csupp 7445  seq𝜔cseqom 7694   +𝑜 coa 7709   ·𝑜 comu 7710   ↑𝑜 coe 7711   finSupp cfsupp 8430  OrdIsocoi 8569   CNF ccnf 8721 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-supp 7446  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-seqom 7695  df-1o 7712  df-2o 7713  df-oadd 7716  df-omul 7717  df-oexp 7718  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-fsupp 8431  df-oi 8570  df-cnf 8722 This theorem is referenced by:  cnfcom2  8762
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