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Theorem cnfcom 8874
 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 8820 . . . . . . 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 8870 . . . . . . . . 9 (𝜑 → (ω CNF 𝐴):𝑆1-1-onto→(ω ↑o 𝐴))
9 f1ocnv 6390 . . . . . . . . 9 ((ω CNF 𝐴):𝑆1-1-onto→(ω ↑o 𝐴) → (ω CNF 𝐴):(ω ↑o 𝐴)–1-1-onto𝑆)
10 f1of 6378 . . . . . . . . 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 6609 . . . . . . 7 (𝜑 → ((ω CNF 𝐴)‘𝐵) ∈ 𝑆)
147, 13syl5eqel 2910 . . . . . 6 (𝜑𝐹𝑆)
152, 4, 5, 6, 14cantnfcl 8841 . . . . 5 (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
1615simprd 491 . . . 4 (𝜑 → dom 𝐺 ∈ ω)
17 elnn 7336 . . . 4 ((𝐼 ∈ dom 𝐺 ∧ dom 𝐺 ∈ ω) → 𝐼 ∈ ω)
181, 16, 17syl2anc 581 . . 3 (𝜑𝐼 ∈ ω)
19 eleq1 2894 . . . . . 6 (𝑤 = 𝐼 → (𝑤 ∈ dom 𝐺𝐼 ∈ dom 𝐺))
20 suceq 6028 . . . . . . . 8 (𝑤 = 𝐼 → suc 𝑤 = suc 𝐼)
2120fveq2d 6437 . . . . . . 7 (𝑤 = 𝐼 → (𝑇‘suc 𝑤) = (𝑇‘suc 𝐼))
2220fveq2d 6437 . . . . . . 7 (𝑤 = 𝐼 → (𝐻‘suc 𝑤) = (𝐻‘suc 𝐼))
23 fveq2 6433 . . . . . . . . 9 (𝑤 = 𝐼 → (𝐺𝑤) = (𝐺𝐼))
2423oveq2d 6921 . . . . . . . 8 (𝑤 = 𝐼 → (ω ↑o (𝐺𝑤)) = (ω ↑o (𝐺𝐼)))
25 2fveq3 6438 . . . . . . . 8 (𝑤 = 𝐼 → (𝐹‘(𝐺𝑤)) = (𝐹‘(𝐺𝐼)))
2624, 25oveq12d 6923 . . . . . . 7 (𝑤 = 𝐼 → ((ω ↑o (𝐺𝑤)) ·o (𝐹‘(𝐺𝑤))) = ((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))))
2721, 22, 26f1oeq123d 6373 . . . . . 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 336 . . . . 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 332 . . . 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 2894 . . . . . 6 (𝑤 = ∅ → (𝑤 ∈ dom 𝐺 ↔ ∅ ∈ dom 𝐺))
31 suceq 6028 . . . . . . . 8 (𝑤 = ∅ → suc 𝑤 = suc ∅)
3231fveq2d 6437 . . . . . . 7 (𝑤 = ∅ → (𝑇‘suc 𝑤) = (𝑇‘suc ∅))
3331fveq2d 6437 . . . . . . 7 (𝑤 = ∅ → (𝐻‘suc 𝑤) = (𝐻‘suc ∅))
34 fveq2 6433 . . . . . . . . 9 (𝑤 = ∅ → (𝐺𝑤) = (𝐺‘∅))
3534oveq2d 6921 . . . . . . . 8 (𝑤 = ∅ → (ω ↑o (𝐺𝑤)) = (ω ↑o (𝐺‘∅)))
36 2fveq3 6438 . . . . . . . 8 (𝑤 = ∅ → (𝐹‘(𝐺𝑤)) = (𝐹‘(𝐺‘∅)))
3735, 36oveq12d 6923 . . . . . . 7 (𝑤 = ∅ → ((ω ↑o (𝐺𝑤)) ·o (𝐹‘(𝐺𝑤))) = ((ω ↑o (𝐺‘∅)) ·o (𝐹‘(𝐺‘∅))))
3832, 33, 37f1oeq123d 6373 . . . . . 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 336 . . . . 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 2894 . . . . . 6 (𝑤 = 𝑦 → (𝑤 ∈ dom 𝐺𝑦 ∈ dom 𝐺))
41 suceq 6028 . . . . . . . 8 (𝑤 = 𝑦 → suc 𝑤 = suc 𝑦)
4241fveq2d 6437 . . . . . . 7 (𝑤 = 𝑦 → (𝑇‘suc 𝑤) = (𝑇‘suc 𝑦))
4341fveq2d 6437 . . . . . . 7 (𝑤 = 𝑦 → (𝐻‘suc 𝑤) = (𝐻‘suc 𝑦))
44 fveq2 6433 . . . . . . . . 9 (𝑤 = 𝑦 → (𝐺𝑤) = (𝐺𝑦))
4544oveq2d 6921 . . . . . . . 8 (𝑤 = 𝑦 → (ω ↑o (𝐺𝑤)) = (ω ↑o (𝐺𝑦)))
46 2fveq3 6438 . . . . . . . 8 (𝑤 = 𝑦 → (𝐹‘(𝐺𝑤)) = (𝐹‘(𝐺𝑦)))
4745, 46oveq12d 6923 . . . . . . 7 (𝑤 = 𝑦 → ((ω ↑o (𝐺𝑤)) ·o (𝐹‘(𝐺𝑤))) = ((ω ↑o (𝐺𝑦)) ·o (𝐹‘(𝐺𝑦))))
4842, 43, 47f1oeq123d 6373 . . . . . 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 336 . . . . 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 2894 . . . . . 6 (𝑤 = suc 𝑦 → (𝑤 ∈ dom 𝐺 ↔ suc 𝑦 ∈ dom 𝐺))
51 suceq 6028 . . . . . . . 8 (𝑤 = suc 𝑦 → suc 𝑤 = suc suc 𝑦)
5251fveq2d 6437 . . . . . . 7 (𝑤 = suc 𝑦 → (𝑇‘suc 𝑤) = (𝑇‘suc suc 𝑦))
5351fveq2d 6437 . . . . . . 7 (𝑤 = suc 𝑦 → (𝐻‘suc 𝑤) = (𝐻‘suc suc 𝑦))
54 fveq2 6433 . . . . . . . . 9 (𝑤 = suc 𝑦 → (𝐺𝑤) = (𝐺‘suc 𝑦))
5554oveq2d 6921 . . . . . . . 8 (𝑤 = suc 𝑦 → (ω ↑o (𝐺𝑤)) = (ω ↑o (𝐺‘suc 𝑦)))
56 2fveq3 6438 . . . . . . . 8 (𝑤 = suc 𝑦 → (𝐹‘(𝐺𝑤)) = (𝐹‘(𝐺‘suc 𝑦)))
5755, 56oveq12d 6923 . . . . . . 7 (𝑤 = suc 𝑦 → ((ω ↑o (𝐺𝑤)) ·o (𝐹‘(𝐺𝑤))) = ((ω ↑o (𝐺‘suc 𝑦)) ·o (𝐹‘(𝐺‘suc 𝑦))))
5852, 53, 57f1oeq123d 6373 . . . . . 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 336 . . . . 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 474 . . . . . . 7 ((𝜑 ∧ ∅ ∈ dom 𝐺) → 𝐴 ∈ On)
6112adantr 474 . . . . . . 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 479 . . . . . . 7 ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ dom 𝐺)
673a1i 11 . . . . . . . 8 ((𝜑 ∧ ∅ ∈ dom 𝐺) → ω ∈ On)
68 suppssdm 7572 . . . . . . . . . . 11 (𝐹 supp ∅) ⊆ dom 𝐹
692, 4, 5cantnfs 8840 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
7014, 69mpbid 224 . . . . . . . . . . . 12 (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
7170simpld 490 . . . . . . . . . . 11 (𝜑𝐹:𝐴⟶ω)
7268, 71fssdm 6294 . . . . . . . . . 10 (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
73 onss 7251 . . . . . . . . . . 11 (𝐴 ∈ On → 𝐴 ⊆ On)
745, 73syl 17 . . . . . . . . . 10 (𝜑𝐴 ⊆ On)
7572, 74sstrd 3837 . . . . . . . . 9 (𝜑 → (𝐹 supp ∅) ⊆ On)
766oif 8704 . . . . . . . . . 10 𝐺:dom 𝐺⟶(𝐹 supp ∅)
7776ffvelrni 6607 . . . . . . . . 9 (∅ ∈ dom 𝐺 → (𝐺‘∅) ∈ (𝐹 supp ∅))
78 ssel2 3822 . . . . . . . . 9 (((𝐹 supp ∅) ⊆ On ∧ (𝐺‘∅) ∈ (𝐹 supp ∅)) → (𝐺‘∅) ∈ On)
7975, 77, 78syl2an 591 . . . . . . . 8 ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝐺‘∅) ∈ On)
80 peano1 7346 . . . . . . . . 9 ∅ ∈ ω
8180a1i 11 . . . . . . . 8 ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ ω)
82 oen0 7933 . . . . . . . 8 (((ω ∈ On ∧ (𝐺‘∅) ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o (𝐺‘∅)))
8367, 79, 81, 82syl21anc 873 . . . . . . 7 ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ (ω ↑o (𝐺‘∅)))
84 0ex 5014 . . . . . . . . 9 ∅ ∈ V
8563seqom0g 7817 . . . . . . . . 9 (∅ ∈ V → (𝑇‘∅) = ∅)
8684, 85ax-mp 5 . . . . . . . 8 (𝑇‘∅) = ∅
87 f1o0 6414 . . . . . . . . . 10 ∅:∅–1-1-onto→∅
8862seqom0g 7817 . . . . . . . . . . 11 (∅ ∈ V → (𝐻‘∅) = ∅)
89 f1oeq2 6368 . . . . . . . . . . 11 ((𝐻‘∅) = ∅ → (∅:(𝐻‘∅)–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
9084, 88, 89mp2b 10 . . . . . . . . . 10 (∅:(𝐻‘∅)–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅)
9187, 90mpbir 223 . . . . . . . . 9 ∅:(𝐻‘∅)–1-1-onto→∅
92 f1oeq1 6367 . . . . . . . . 9 ((𝑇‘∅) = ∅ → ((𝑇‘∅):(𝐻‘∅)–1-1-onto→∅ ↔ ∅:(𝐻‘∅)–1-1-onto→∅))
9391, 92mpbiri 250 . . . . . . . 8 ((𝑇‘∅) = ∅ → (𝑇‘∅):(𝐻‘∅)–1-1-onto→∅)
9486, 93mp1i 13 . . . . . . 7 ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝑇‘∅):(𝐻‘∅)–1-1-onto→∅)
952, 60, 61, 7, 6, 62, 63, 64, 65, 66, 83, 94cnfcomlem 8873 . . . . . 6 ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑o (𝐺‘∅)) ·o (𝐹‘(𝐺‘∅))))
9695ex 403 . . . . 5 (𝜑 → (∅ ∈ dom 𝐺 → (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑o (𝐺‘∅)) ·o (𝐹‘(𝐺‘∅)))))
976oicl 8703 . . . . . . . . . 10 Ord dom 𝐺
98 ordtr 5977 . . . . . . . . . 10 (Ord dom 𝐺 → Tr dom 𝐺)
9997, 98ax-mp 5 . . . . . . . . 9 Tr dom 𝐺
100 trsuc 6047 . . . . . . . . 9 ((Tr dom 𝐺 ∧ suc 𝑦 ∈ dom 𝐺) → 𝑦 ∈ dom 𝐺)
10199, 100mpan 683 . . . . . . . 8 (suc 𝑦 ∈ dom 𝐺𝑦 ∈ dom 𝐺)
102101imim1i 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)))))
1035ad2antrr 719 . . . . . . . . . 10 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺𝑦)) ·o (𝐹‘(𝐺𝑦))))) → 𝐴 ∈ On)
10412ad2antrr 719 . . . . . . . . . 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))
105 simprl 789 . . . . . . . . . 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)
10674ad2antrr 719 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺𝑦)) ·o (𝐹‘(𝐺𝑦))))) → 𝐴 ⊆ On)
10772ad2antrr 719 . . . . . . . . . . . . . . . 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)
10876ffvelrni 6607 . . . . . . . . . . . . . . . . 17 (suc 𝑦 ∈ dom 𝐺 → (𝐺‘suc 𝑦) ∈ (𝐹 supp ∅))
109108ad2antrl 721 . . . . . . . . . . . . . . . 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 ∅))
110107, 109sseldd 3828 . . . . . . . . . . . . . . 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)
111106, 110sseldd 3828 . . . . . . . . . . . . . 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)
112 eloni 5973 . . . . . . . . . . . . . 14 ((𝐺‘suc 𝑦) ∈ On → Ord (𝐺‘suc 𝑦))
113111, 112syl 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))
114 vex 3417 . . . . . . . . . . . . . . 15 𝑦 ∈ V
115114sucid 6042 . . . . . . . . . . . . . 14 𝑦 ∈ suc 𝑦
1165, 72ssexd 5030 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹 supp ∅) ∈ V)
11715simpld 490 . . . . . . . . . . . . . . . . . 18 (𝜑 → E We (𝐹 supp ∅))
1186oiiso 8711 . . . . . . . . . . . . . . . . . 18 (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
119116, 117, 118syl2anc 581 . . . . . . . . . . . . . . . . 17 (𝜑𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
120119ad2antrr 719 . . . . . . . . . . . . . . . 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 ∅)))
121101ad2antrl 721 . . . . . . . . . . . . . . . 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)
122 isorel 6831 . . . . . . . . . . . . . . . 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)))
123120, 121, 105, 122syl12anc 872 . . . . . . . . . . . . . . 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)))
124114sucex 7272 . . . . . . . . . . . . . . . 16 suc 𝑦 ∈ V
125124epeli 5257 . . . . . . . . . . . . . . 15 (𝑦 E suc 𝑦𝑦 ∈ suc 𝑦)
126 fvex 6446 . . . . . . . . . . . . . . . 16 (𝐺‘suc 𝑦) ∈ V
127126epeli 5257 . . . . . . . . . . . . . . 15 ((𝐺𝑦) E (𝐺‘suc 𝑦) ↔ (𝐺𝑦) ∈ (𝐺‘suc 𝑦))
128123, 125, 1273bitr3g 305 . . . . . . . . . . . . . 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)))
129115, 128mpbii 225 . . . . . . . . . . . . 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))
130 ordsucss 7279 . . . . . . . . . . . . 13 (Ord (𝐺‘suc 𝑦) → ((𝐺𝑦) ∈ (𝐺‘suc 𝑦) → suc (𝐺𝑦) ⊆ (𝐺‘suc 𝑦)))
131113, 129, 130sylc 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))
13276ffvelrni 6607 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ dom 𝐺 → (𝐺𝑦) ∈ (𝐹 supp ∅))
133121, 132syl 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 ∅))
134107, 133sseldd 3828 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺𝑦)) ·o (𝐹‘(𝐺𝑦))))) → (𝐺𝑦) ∈ 𝐴)
135106, 134sseldd 3828 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺𝑦)) ·o (𝐹‘(𝐺𝑦))))) → (𝐺𝑦) ∈ On)
136 suceloni 7274 . . . . . . . . . . . . . 14 ((𝐺𝑦) ∈ On → suc (𝐺𝑦) ∈ On)
137135, 136syl 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)
1383a1i 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)
13980a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ ω) ∧ (suc 𝑦 ∈ dom 𝐺 ∧ (𝑇‘suc 𝑦):(𝐻‘suc 𝑦)–1-1-onto→((ω ↑o (𝐺𝑦)) ·o (𝐹‘(𝐺𝑦))))) → ∅ ∈ ω)
140 oewordi 7938 . . . . . . . . . . . . 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))))
141137, 111, 138, 139, 140syl31anc 1498 . . . . . . . . . . . 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))))
142131, 141mpd 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)))
14371ad2antrr 719 . . . . . . . . . . . . . 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)
144143, 134ffvelrnd 6609 . . . . . . . . . . . . 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)
145 nnon 7332 . . . . . . . . . . . . . . 15 ((𝐹‘(𝐺𝑦)) ∈ ω → (𝐹‘(𝐺𝑦)) ∈ On)
146144, 145syl 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)
147 oecl 7884 . . . . . . . . . . . . . . 15 ((ω ∈ On ∧ (𝐺𝑦) ∈ On) → (ω ↑o (𝐺𝑦)) ∈ On)
148138, 135, 147syl2anc 581 . . . . . . . . . . . . . 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)
149 oen0 7933 . . . . . . . . . . . . . . 15 (((ω ∈ On ∧ (𝐺𝑦) ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o (𝐺𝑦)))
150138, 135, 139, 149syl21anc 873 . . . . . . . . . . . . . 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)))
151 omord2 7914 . . . . . . . . . . . . . 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)))
152146, 138, 148, 150, 151syl31anc 1498 . . . . . . . . . . . . 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)))
153144, 152mpbid 224 . . . . . . . . . . . 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))
154 oesuc 7874 . . . . . . . . . . . . 13 ((ω ∈ On ∧ (𝐺𝑦) ∈ On) → (ω ↑o suc (𝐺𝑦)) = ((ω ↑o (𝐺𝑦)) ·o ω))
155138, 135, 154syl2anc 581 . . . . . . . . . . . 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))
156153, 155eleqtrrd 2909 . . . . . . . . . . 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)))
157142, 156sseldd 3828 . . . . . . . . . 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)))
158 simprr 791 . . . . . . . . . 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))))
1592, 103, 104, 7, 6, 62, 63, 64, 65, 105, 157, 158cnfcomlem 8873 . . . . . . . . 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))))
160159exp32 413 . . . . . . . 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))))))
161160a2d 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))))))
162102, 161syl5 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))))))
163162expcom 404 . . . . 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)))))))
16439, 49, 59, 96, 163finds2 7355 . . . 4 (𝑤 ∈ ω → (𝜑 → (𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑o (𝐺𝑤)) ·o (𝐹‘(𝐺𝑤))))))
16529, 164vtoclga 3489 . . 3 (𝐼 ∈ ω → (𝜑 → (𝐼 ∈ dom 𝐺 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))))))
16618, 165mpcom 38 . 2 (𝜑 → (𝐼 ∈ dom 𝐺 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼)))))
1671, 166mpd 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 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  Vcvv 3414   ∪ cun 3796   ⊆ wss 3798  ∅c0 4144   class class class wbr 4873   ↦ cmpt 4952  Tr wtr 4975   E cep 5254   We wwe 5300  ◡ccnv 5341  dom cdm 5342  Ord word 5962  Oncon0 5963  suc csuc 5965  ⟶wf 6119  –1-1-onto→wf1o 6122  ‘cfv 6123   Isom wiso 6124  (class class class)co 6905   ↦ cmpt2 6907  ωcom 7326   supp csupp 7559  seq𝜔cseqom 7808   +o coa 7823   ·o comu 7824   ↑o coe 7825   finSupp cfsupp 8544  OrdIsocoi 8683   CNF ccnf 8835 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-inf2 8815 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-supp 7560  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-seqom 7809  df-1o 7826  df-2o 7827  df-oadd 7830  df-omul 7831  df-oexp 7832  df-er 8009  df-map 8124  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-fsupp 8545  df-oi 8684  df-cnf 8836 This theorem is referenced by:  cnfcom2  8876
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