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Theorem cnfcom 8480
 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 8426 . . . . . . 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 8476 . . . . . . . . 9 (𝜑 → (ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴))
9 f1ocnv 6062 . . . . . . . . 9 ((ω CNF 𝐴):𝑆1-1-onto→(ω ↑𝑜 𝐴) → (ω CNF 𝐴):(ω ↑𝑜 𝐴)–1-1-onto𝑆)
10 f1of 6050 . . . . . . . . 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 6268 . . . . . . 7 (𝜑 → ((ω CNF 𝐴)‘𝐵) ∈ 𝑆)
147, 13syl5eqel 2692 . . . . . 6 (𝜑𝐹𝑆)
152, 4, 5, 6, 14cantnfcl 8447 . . . . 5 (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
1615simprd 478 . . . 4 (𝜑 → dom 𝐺 ∈ ω)
17 elnn 6967 . . . 4 ((𝐼 ∈ dom 𝐺 ∧ dom 𝐺 ∈ ω) → 𝐼 ∈ ω)
181, 16, 17syl2anc 691 . . 3 (𝜑𝐼 ∈ ω)
19 eleq1 2676 . . . . . 6 (𝑤 = 𝐼 → (𝑤 ∈ dom 𝐺𝐼 ∈ dom 𝐺))
20 suceq 5707 . . . . . . . 8 (𝑤 = 𝐼 → suc 𝑤 = suc 𝐼)
2120fveq2d 6107 . . . . . . 7 (𝑤 = 𝐼 → (𝑇‘suc 𝑤) = (𝑇‘suc 𝐼))
2220fveq2d 6107 . . . . . . 7 (𝑤 = 𝐼 → (𝐻‘suc 𝑤) = (𝐻‘suc 𝐼))
23 fveq2 6103 . . . . . . . . 9 (𝑤 = 𝐼 → (𝐺𝑤) = (𝐺𝐼))
2423oveq2d 6565 . . . . . . . 8 (𝑤 = 𝐼 → (ω ↑𝑜 (𝐺𝑤)) = (ω ↑𝑜 (𝐺𝐼)))
2523fveq2d 6107 . . . . . . . 8 (𝑤 = 𝐼 → (𝐹‘(𝐺𝑤)) = (𝐹‘(𝐺𝐼)))
2624, 25oveq12d 6567 . . . . . . 7 (𝑤 = 𝐼 → ((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤))) = ((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))
2721, 22, 26f1oeq123d 6046 . . . . . 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 2676 . . . . . 6 (𝑤 = ∅ → (𝑤 ∈ dom 𝐺 ↔ ∅ ∈ dom 𝐺))
31 suceq 5707 . . . . . . . 8 (𝑤 = ∅ → suc 𝑤 = suc ∅)
3231fveq2d 6107 . . . . . . 7 (𝑤 = ∅ → (𝑇‘suc 𝑤) = (𝑇‘suc ∅))
3331fveq2d 6107 . . . . . . 7 (𝑤 = ∅ → (𝐻‘suc 𝑤) = (𝐻‘suc ∅))
34 fveq2 6103 . . . . . . . . 9 (𝑤 = ∅ → (𝐺𝑤) = (𝐺‘∅))
3534oveq2d 6565 . . . . . . . 8 (𝑤 = ∅ → (ω ↑𝑜 (𝐺𝑤)) = (ω ↑𝑜 (𝐺‘∅)))
3634fveq2d 6107 . . . . . . . 8 (𝑤 = ∅ → (𝐹‘(𝐺𝑤)) = (𝐹‘(𝐺‘∅)))
3735, 36oveq12d 6567 . . . . . . 7 (𝑤 = ∅ → ((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤))) = ((ω ↑𝑜 (𝐺‘∅)) ·𝑜 (𝐹‘(𝐺‘∅))))
3832, 33, 37f1oeq123d 6046 . . . . . 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 2676 . . . . . 6 (𝑤 = 𝑦 → (𝑤 ∈ dom 𝐺𝑦 ∈ dom 𝐺))
41 suceq 5707 . . . . . . . 8 (𝑤 = 𝑦 → suc 𝑤 = suc 𝑦)
4241fveq2d 6107 . . . . . . 7 (𝑤 = 𝑦 → (𝑇‘suc 𝑤) = (𝑇‘suc 𝑦))
4341fveq2d 6107 . . . . . . 7 (𝑤 = 𝑦 → (𝐻‘suc 𝑤) = (𝐻‘suc 𝑦))
44 fveq2 6103 . . . . . . . . 9 (𝑤 = 𝑦 → (𝐺𝑤) = (𝐺𝑦))
4544oveq2d 6565 . . . . . . . 8 (𝑤 = 𝑦 → (ω ↑𝑜 (𝐺𝑤)) = (ω ↑𝑜 (𝐺𝑦)))
4644fveq2d 6107 . . . . . . . 8 (𝑤 = 𝑦 → (𝐹‘(𝐺𝑤)) = (𝐹‘(𝐺𝑦)))
4745, 46oveq12d 6567 . . . . . . 7 (𝑤 = 𝑦 → ((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤))) = ((ω ↑𝑜 (𝐺𝑦)) ·𝑜 (𝐹‘(𝐺𝑦))))
4842, 43, 47f1oeq123d 6046 . . . . . 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 2676 . . . . . 6 (𝑤 = suc 𝑦 → (𝑤 ∈ dom 𝐺 ↔ suc 𝑦 ∈ dom 𝐺))
51 suceq 5707 . . . . . . . 8 (𝑤 = suc 𝑦 → suc 𝑤 = suc suc 𝑦)
5251fveq2d 6107 . . . . . . 7 (𝑤 = suc 𝑦 → (𝑇‘suc 𝑤) = (𝑇‘suc suc 𝑦))
5351fveq2d 6107 . . . . . . 7 (𝑤 = suc 𝑦 → (𝐻‘suc 𝑤) = (𝐻‘suc suc 𝑦))
54 fveq2 6103 . . . . . . . . 9 (𝑤 = suc 𝑦 → (𝐺𝑤) = (𝐺‘suc 𝑦))
5554oveq2d 6565 . . . . . . . 8 (𝑤 = suc 𝑦 → (ω ↑𝑜 (𝐺𝑤)) = (ω ↑𝑜 (𝐺‘suc 𝑦)))
5654fveq2d 6107 . . . . . . . 8 (𝑤 = suc 𝑦 → (𝐹‘(𝐺𝑤)) = (𝐹‘(𝐺‘suc 𝑦)))
5755, 56oveq12d 6567 . . . . . . 7 (𝑤 = suc 𝑦 → ((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤))) = ((ω ↑𝑜 (𝐺‘suc 𝑦)) ·𝑜 (𝐹‘(𝐺‘suc 𝑦))))
5852, 53, 57f1oeq123d 6046 . . . . . 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 480 . . . . . . 7 ((𝜑 ∧ ∅ ∈ dom 𝐺) → 𝐴 ∈ On)
6112adantr 480 . . . . . . 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 476 . . . . . . 7 ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ dom 𝐺)
673a1i 11 . . . . . . . 8 ((𝜑 ∧ ∅ ∈ dom 𝐺) → ω ∈ On)
68 suppssdm 7195 . . . . . . . . . . 11 (𝐹 supp ∅) ⊆ dom 𝐹
692, 4, 5cantnfs 8446 . . . . . . . . . . . . . 14 (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
7014, 69mpbid 221 . . . . . . . . . . . . 13 (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
7170simpld 474 . . . . . . . . . . . 12 (𝜑𝐹:𝐴⟶ω)
72 fdm 5964 . . . . . . . . . . . 12 (𝐹:𝐴⟶ω → dom 𝐹 = 𝐴)
7371, 72syl 17 . . . . . . . . . . 11 (𝜑 → dom 𝐹 = 𝐴)
7468, 73syl5sseq 3616 . . . . . . . . . 10 (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
75 onss 6882 . . . . . . . . . . 11 (𝐴 ∈ On → 𝐴 ⊆ On)
765, 75syl 17 . . . . . . . . . 10 (𝜑𝐴 ⊆ On)
7774, 76sstrd 3578 . . . . . . . . 9 (𝜑 → (𝐹 supp ∅) ⊆ On)
786oif 8318 . . . . . . . . . 10 𝐺:dom 𝐺⟶(𝐹 supp ∅)
7978ffvelrni 6266 . . . . . . . . 9 (∅ ∈ dom 𝐺 → (𝐺‘∅) ∈ (𝐹 supp ∅))
80 ssel2 3563 . . . . . . . . 9 (((𝐹 supp ∅) ⊆ On ∧ (𝐺‘∅) ∈ (𝐹 supp ∅)) → (𝐺‘∅) ∈ On)
8177, 79, 80syl2an 493 . . . . . . . 8 ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝐺‘∅) ∈ On)
82 peano1 6977 . . . . . . . . 9 ∅ ∈ ω
8382a1i 11 . . . . . . . 8 ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ ω)
84 oen0 7553 . . . . . . . 8 (((ω ∈ On ∧ (𝐺‘∅) ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑𝑜 (𝐺‘∅)))
8567, 81, 83, 84syl21anc 1317 . . . . . . 7 ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ (ω ↑𝑜 (𝐺‘∅)))
86 0ex 4718 . . . . . . . . 9 ∅ ∈ V
8763seqom0g 7438 . . . . . . . . 9 (∅ ∈ V → (𝑇‘∅) = ∅)
8886, 87ax-mp 5 . . . . . . . 8 (𝑇‘∅) = ∅
89 f1o0 6085 . . . . . . . . . 10 ∅:∅–1-1-onto→∅
9062seqom0g 7438 . . . . . . . . . . 11 (∅ ∈ V → (𝐻‘∅) = ∅)
91 f1oeq2 6041 . . . . . . . . . . 11 ((𝐻‘∅) = ∅ → (∅:(𝐻‘∅)–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
9286, 90, 91mp2b 10 . . . . . . . . . 10 (∅:(𝐻‘∅)–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅)
9389, 92mpbir 220 . . . . . . . . 9 ∅:(𝐻‘∅)–1-1-onto→∅
94 f1oeq1 6040 . . . . . . . . 9 ((𝑇‘∅) = ∅ → ((𝑇‘∅):(𝐻‘∅)–1-1-onto→∅ ↔ ∅:(𝐻‘∅)–1-1-onto→∅))
9593, 94mpbiri 247 . . . . . . . 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 8479 . . . . . 6 ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑𝑜 (𝐺‘∅)) ·𝑜 (𝐹‘(𝐺‘∅))))
9897ex 449 . . . . 5 (𝜑 → (∅ ∈ dom 𝐺 → (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑𝑜 (𝐺‘∅)) ·𝑜 (𝐹‘(𝐺‘∅)))))
996oicl 8317 . . . . . . . . . 10 Ord dom 𝐺
100 ordtr 5654 . . . . . . . . . 10 (Ord dom 𝐺 → Tr dom 𝐺)
10199, 100ax-mp 5 . . . . . . . . 9 Tr dom 𝐺
102 trsuc 5727 . . . . . . . . 9 ((Tr dom 𝐺 ∧ suc 𝑦 ∈ dom 𝐺) → 𝑦 ∈ dom 𝐺)
103101, 102mpan 702 . . . . . . . 8 (suc 𝑦 ∈ dom 𝐺𝑦 ∈ dom 𝐺)
104103imim1i 61 . . . . . . 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 758 . . . . . . . . . 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 758 . . . . . . . . . 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 790 . . . . . . . . . 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 758 . . . . . . . . . . . . . . 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 758 . . . . . . . . . . . . . . . 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 6266 . . . . . . . . . . . . . . . . 17 (suc 𝑦 ∈ dom 𝐺 → (𝐺‘suc 𝑦) ∈ (𝐹 supp ∅))
111110ad2antrl 760 . . . . . . . . . . . . . . . 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 3569 . . . . . . . . . . . . . . 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 3569 . . . . . . . . . . . . . 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 5650 . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . . . 15 𝑦 ∈ V
117116sucid 5721 . . . . . . . . . . . . . 14 𝑦 ∈ suc 𝑦
1185, 74ssexd 4733 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹 supp ∅) ∈ V)
11915simpld 474 . . . . . . . . . . . . . . . . . 18 (𝜑 → E We (𝐹 supp ∅))
1206oiiso 8325 . . . . . . . . . . . . . . . . . 18 (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
121118, 119, 120syl2anc 691 . . . . . . . . . . . . . . . . 17 (𝜑𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
122121ad2antrr 758 . . . . . . . . . . . . . . . 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 760 . . . . . . . . . . . . . . . 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 6476 . . . . . . . . . . . . . . . 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 1316 . . . . . . . . . . . . . . 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 6903 . . . . . . . . . . . . . . . 16 suc 𝑦 ∈ V
127126epelc 4951 . . . . . . . . . . . . . . 15 (𝑦 E suc 𝑦𝑦 ∈ suc 𝑦)
128 fvex 6113 . . . . . . . . . . . . . . . 16 (𝐺‘suc 𝑦) ∈ V
129128epelc 4951 . . . . . . . . . . . . . . 15 ((𝐺𝑦) E (𝐺‘suc 𝑦) ↔ (𝐺𝑦) ∈ (𝐺‘suc 𝑦))
130125, 127, 1293bitr3g 301 . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . 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 6910 . . . . . . . . . . . . 13 (Ord (𝐺‘suc 𝑦) → ((𝐺𝑦) ∈ (𝐺‘suc 𝑦) → suc (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)))
133115, 131, 132sylc 63 . . . . . . . . . . . 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 6266 . . . . . . . . . . . . . . . . 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 3569 . . . . . . . . . . . . . . 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 3569 . . . . . . . . . . . . . 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 6905 . . . . . . . . . . . . . 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 7558 . . . . . . . . . . . . 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 1321 . . . . . . . . . . . 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 758 . . . . . . . . . . . . . 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 6268 . . . . . . . . . . . . 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 6963 . . . . . . . . . . . . . . 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 7504 . . . . . . . . . . . . . . 15 ((ω ∈ On ∧ (𝐺𝑦) ∈ On) → (ω ↑𝑜 (𝐺𝑦)) ∈ On)
150140, 137, 149syl2anc 691 . . . . . . . . . . . . . 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 7553 . . . . . . . . . . . . . . 15 (((ω ∈ On ∧ (𝐺𝑦) ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑𝑜 (𝐺𝑦)))
152140, 137, 141, 151syl21anc 1317 . . . . . . . . . . . . . 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 7534 . . . . . . . . . . . . . 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 1321 . . . . . . . . . . . . 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 221 . . . . . . . . . . . 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 7494 . . . . . . . . . . . . 13 ((ω ∈ On ∧ (𝐺𝑦) ∈ On) → (ω ↑𝑜 suc (𝐺𝑦)) = ((ω ↑𝑜 (𝐺𝑦)) ·𝑜 ω))
157140, 137, 156syl2anc 691 . . . . . . . . . . . 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 2691 . . . . . . . . . . 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 3569 . . . . . . . . . 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 792 . . . . . . . . . 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 8479 . . . . . . . . 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 629 . . . . . . . 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 33 . . . . . 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 450 . . . . 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 6986 . . . 4 (𝑤 ∈ ω → (𝜑 → (𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑𝑜 (𝐺𝑤)) ·𝑜 (𝐹‘(𝐺𝑤))))))
16729, 166vtoclga 3245 . . 3 (𝐼 ∈ ω → (𝜑 → (𝐼 ∈ dom 𝐺 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑𝑜 (𝐺𝐼)) ·𝑜 (𝐹‘(𝐺𝐼))))))
16818, 167mpcom 37 . 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583   ↦ cmpt 4643  Tr wtr 4680   E cep 4947   We wwe 4996  ◡ccnv 5037  dom cdm 5038  Ord word 5639  Oncon0 5640  suc csuc 5642  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957   supp csupp 7182  seq𝜔cseqom 7429   +𝑜 coa 7444   ·𝑜 comu 7445   ↑𝑜 coe 7446   finSupp cfsupp 8158  OrdIsocoi 8297   CNF ccnf 8441 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seqom 7430  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-oexp 7453  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-cnf 8442 This theorem is referenced by:  cnfcom2  8482
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