MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnfcom Structured version   Visualization version   GIF version

Theorem cnfcom 9157
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 9103 . . . . . . 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 9153 . . . . . . . . 9 (𝜑 → (ω CNF 𝐴):𝑆1-1-onto→(ω ↑o 𝐴))
9 f1ocnv 6621 . . . . . . . . 9 ((ω CNF 𝐴):𝑆1-1-onto→(ω ↑o 𝐴) → (ω CNF 𝐴):(ω ↑o 𝐴)–1-1-onto𝑆)
10 f1of 6609 . . . . . . . . 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 6846 . . . . . . 7 (𝜑 → ((ω CNF 𝐴)‘𝐵) ∈ 𝑆)
147, 13eqeltrid 2917 . . . . . 6 (𝜑𝐹𝑆)
152, 4, 5, 6, 14cantnfcl 9124 . . . . 5 (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
1615simprd 498 . . . 4 (𝜑 → dom 𝐺 ∈ ω)
17 elnn 7584 . . . 4 ((𝐼 ∈ dom 𝐺 ∧ dom 𝐺 ∈ ω) → 𝐼 ∈ ω)
181, 16, 17syl2anc 586 . . 3 (𝜑𝐼 ∈ ω)
19 eleq1 2900 . . . . . 6 (𝑤 = 𝐼 → (𝑤 ∈ dom 𝐺𝐼 ∈ dom 𝐺))
20 suceq 6250 . . . . . . . 8 (𝑤 = 𝐼 → suc 𝑤 = suc 𝐼)
2120fveq2d 6668 . . . . . . 7 (𝑤 = 𝐼 → (𝑇‘suc 𝑤) = (𝑇‘suc 𝐼))
2220fveq2d 6668 . . . . . . 7 (𝑤 = 𝐼 → (𝐻‘suc 𝑤) = (𝐻‘suc 𝐼))
23 fveq2 6664 . . . . . . . . 9 (𝑤 = 𝐼 → (𝐺𝑤) = (𝐺𝐼))
2423oveq2d 7166 . . . . . . . 8 (𝑤 = 𝐼 → (ω ↑o (𝐺𝑤)) = (ω ↑o (𝐺𝐼)))
25 2fveq3 6669 . . . . . . . 8 (𝑤 = 𝐼 → (𝐹‘(𝐺𝑤)) = (𝐹‘(𝐺𝐼)))
2624, 25oveq12d 7168 . . . . . . 7 (𝑤 = 𝐼 → ((ω ↑o (𝐺𝑤)) ·o (𝐹‘(𝐺𝑤))) = ((ω ↑o (𝐺𝐼)) ·o (𝐹‘(𝐺𝐼))))
2721, 22, 26f1oeq123d 6604 . . . . . 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 347 . . . . 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 343 . . . 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 2900 . . . . . 6 (𝑤 = ∅ → (𝑤 ∈ dom 𝐺 ↔ ∅ ∈ dom 𝐺))
31 suceq 6250 . . . . . . . 8 (𝑤 = ∅ → suc 𝑤 = suc ∅)
3231fveq2d 6668 . . . . . . 7 (𝑤 = ∅ → (𝑇‘suc 𝑤) = (𝑇‘suc ∅))
3331fveq2d 6668 . . . . . . 7 (𝑤 = ∅ → (𝐻‘suc 𝑤) = (𝐻‘suc ∅))
34 fveq2 6664 . . . . . . . . 9 (𝑤 = ∅ → (𝐺𝑤) = (𝐺‘∅))
3534oveq2d 7166 . . . . . . . 8 (𝑤 = ∅ → (ω ↑o (𝐺𝑤)) = (ω ↑o (𝐺‘∅)))
36 2fveq3 6669 . . . . . . . 8 (𝑤 = ∅ → (𝐹‘(𝐺𝑤)) = (𝐹‘(𝐺‘∅)))
3735, 36oveq12d 7168 . . . . . . 7 (𝑤 = ∅ → ((ω ↑o (𝐺𝑤)) ·o (𝐹‘(𝐺𝑤))) = ((ω ↑o (𝐺‘∅)) ·o (𝐹‘(𝐺‘∅))))
3832, 33, 37f1oeq123d 6604 . . . . . 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 347 . . . . 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 2900 . . . . . 6 (𝑤 = 𝑦 → (𝑤 ∈ dom 𝐺𝑦 ∈ dom 𝐺))
41 suceq 6250 . . . . . . . 8 (𝑤 = 𝑦 → suc 𝑤 = suc 𝑦)
4241fveq2d 6668 . . . . . . 7 (𝑤 = 𝑦 → (𝑇‘suc 𝑤) = (𝑇‘suc 𝑦))
4341fveq2d 6668 . . . . . . 7 (𝑤 = 𝑦 → (𝐻‘suc 𝑤) = (𝐻‘suc 𝑦))
44 fveq2 6664 . . . . . . . . 9 (𝑤 = 𝑦 → (𝐺𝑤) = (𝐺𝑦))
4544oveq2d 7166 . . . . . . . 8 (𝑤 = 𝑦 → (ω ↑o (𝐺𝑤)) = (ω ↑o (𝐺𝑦)))
46 2fveq3 6669 . . . . . . . 8 (𝑤 = 𝑦 → (𝐹‘(𝐺𝑤)) = (𝐹‘(𝐺𝑦)))
4745, 46oveq12d 7168 . . . . . . 7 (𝑤 = 𝑦 → ((ω ↑o (𝐺𝑤)) ·o (𝐹‘(𝐺𝑤))) = ((ω ↑o (𝐺𝑦)) ·o (𝐹‘(𝐺𝑦))))
4842, 43, 47f1oeq123d 6604 . . . . . 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 347 . . . . 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 2900 . . . . . 6 (𝑤 = suc 𝑦 → (𝑤 ∈ dom 𝐺 ↔ suc 𝑦 ∈ dom 𝐺))
51 suceq 6250 . . . . . . . 8 (𝑤 = suc 𝑦 → suc 𝑤 = suc suc 𝑦)
5251fveq2d 6668 . . . . . . 7 (𝑤 = suc 𝑦 → (𝑇‘suc 𝑤) = (𝑇‘suc suc 𝑦))
5351fveq2d 6668 . . . . . . 7 (𝑤 = suc 𝑦 → (𝐻‘suc 𝑤) = (𝐻‘suc suc 𝑦))
54 fveq2 6664 . . . . . . . . 9 (𝑤 = suc 𝑦 → (𝐺𝑤) = (𝐺‘suc 𝑦))
5554oveq2d 7166 . . . . . . . 8 (𝑤 = suc 𝑦 → (ω ↑o (𝐺𝑤)) = (ω ↑o (𝐺‘suc 𝑦)))
56 2fveq3 6669 . . . . . . . 8 (𝑤 = suc 𝑦 → (𝐹‘(𝐺𝑤)) = (𝐹‘(𝐺‘suc 𝑦)))
5755, 56oveq12d 7168 . . . . . . 7 (𝑤 = suc 𝑦 → ((ω ↑o (𝐺𝑤)) ·o (𝐹‘(𝐺𝑤))) = ((ω ↑o (𝐺‘suc 𝑦)) ·o (𝐹‘(𝐺‘suc 𝑦))))
5852, 53, 57f1oeq123d 6604 . . . . . 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 347 . . . . 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 483 . . . . . . 7 ((𝜑 ∧ ∅ ∈ dom 𝐺) → 𝐴 ∈ On)
6112adantr 483 . . . . . . 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 487 . . . . . . 7 ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ dom 𝐺)
673a1i 11 . . . . . . . 8 ((𝜑 ∧ ∅ ∈ dom 𝐺) → ω ∈ On)
68 suppssdm 7837 . . . . . . . . . . 11 (𝐹 supp ∅) ⊆ dom 𝐹
692, 4, 5cantnfs 9123 . . . . . . . . . . . . 13 (𝜑 → (𝐹𝑆 ↔ (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅)))
7014, 69mpbid 234 . . . . . . . . . . . 12 (𝜑 → (𝐹:𝐴⟶ω ∧ 𝐹 finSupp ∅))
7170simpld 497 . . . . . . . . . . 11 (𝜑𝐹:𝐴⟶ω)
7268, 71fssdm 6524 . . . . . . . . . 10 (𝜑 → (𝐹 supp ∅) ⊆ 𝐴)
73 onss 7499 . . . . . . . . . . 11 (𝐴 ∈ On → 𝐴 ⊆ On)
745, 73syl 17 . . . . . . . . . 10 (𝜑𝐴 ⊆ On)
7572, 74sstrd 3976 . . . . . . . . 9 (𝜑 → (𝐹 supp ∅) ⊆ On)
766oif 8988 . . . . . . . . . 10 𝐺:dom 𝐺⟶(𝐹 supp ∅)
7776ffvelrni 6844 . . . . . . . . 9 (∅ ∈ dom 𝐺 → (𝐺‘∅) ∈ (𝐹 supp ∅))
78 ssel2 3961 . . . . . . . . 9 (((𝐹 supp ∅) ⊆ On ∧ (𝐺‘∅) ∈ (𝐹 supp ∅)) → (𝐺‘∅) ∈ On)
7975, 77, 78syl2an 597 . . . . . . . 8 ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝐺‘∅) ∈ On)
80 peano1 7595 . . . . . . . . 9 ∅ ∈ ω
8180a1i 11 . . . . . . . 8 ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ ω)
82 oen0 8206 . . . . . . . 8 (((ω ∈ On ∧ (𝐺‘∅) ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o (𝐺‘∅)))
8367, 79, 81, 82syl21anc 835 . . . . . . 7 ((𝜑 ∧ ∅ ∈ dom 𝐺) → ∅ ∈ (ω ↑o (𝐺‘∅)))
84 0ex 5203 . . . . . . . . 9 ∅ ∈ V
8563seqom0g 8086 . . . . . . . . 9 (∅ ∈ V → (𝑇‘∅) = ∅)
8684, 85ax-mp 5 . . . . . . . 8 (𝑇‘∅) = ∅
87 f1o0 6645 . . . . . . . . . 10 ∅:∅–1-1-onto→∅
8862seqom0g 8086 . . . . . . . . . . 11 (∅ ∈ V → (𝐻‘∅) = ∅)
89 f1oeq2 6599 . . . . . . . . . . 11 ((𝐻‘∅) = ∅ → (∅:(𝐻‘∅)–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
9084, 88, 89mp2b 10 . . . . . . . . . 10 (∅:(𝐻‘∅)–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅)
9187, 90mpbir 233 . . . . . . . . 9 ∅:(𝐻‘∅)–1-1-onto→∅
92 f1oeq1 6598 . . . . . . . . 9 ((𝑇‘∅) = ∅ → ((𝑇‘∅):(𝐻‘∅)–1-1-onto→∅ ↔ ∅:(𝐻‘∅)–1-1-onto→∅))
9391, 92mpbiri 260 . . . . . . . 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 9156 . . . . . 6 ((𝜑 ∧ ∅ ∈ dom 𝐺) → (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑o (𝐺‘∅)) ·o (𝐹‘(𝐺‘∅))))
9695ex 415 . . . . 5 (𝜑 → (∅ ∈ dom 𝐺 → (𝑇‘suc ∅):(𝐻‘suc ∅)–1-1-onto→((ω ↑o (𝐺‘∅)) ·o (𝐹‘(𝐺‘∅)))))
976oicl 8987 . . . . . . . . . 10 Ord dom 𝐺
98 ordtr 6199 . . . . . . . . . 10 (Ord dom 𝐺 → Tr dom 𝐺)
9997, 98ax-mp 5 . . . . . . . . 9 Tr dom 𝐺
100 trsuc 6269 . . . . . . . . 9 ((Tr dom 𝐺 ∧ suc 𝑦 ∈ dom 𝐺) → 𝑦 ∈ dom 𝐺)
10199, 100mpan 688 . . . . . . . 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 724 . . . . . . . . . 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 724 . . . . . . . . . 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 769 . . . . . . . . . 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 724 . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . 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 6844 . . . . . . . . . . . . . . . . 17 (suc 𝑦 ∈ dom 𝐺 → (𝐺‘suc 𝑦) ∈ (𝐹 supp ∅))
109108ad2antrl 726 . . . . . . . . . . . . . . . 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 3967 . . . . . . . . . . . . . . 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 3967 . . . . . . . . . . . . . 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 6195 . . . . . . . . . . . . . 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 3497 . . . . . . . . . . . . . . 15 𝑦 ∈ V
115114sucid 6264 . . . . . . . . . . . . . 14 𝑦 ∈ suc 𝑦
116 ovexd 7185 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐹 supp ∅) ∈ V)
11715simpld 497 . . . . . . . . . . . . . . . . . 18 (𝜑 → E We (𝐹 supp ∅))
1186oiiso 8995 . . . . . . . . . . . . . . . . . 18 (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → 𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
119116, 117, 118syl2anc 586 . . . . . . . . . . . . . . . . 17 (𝜑𝐺 Isom E , E (dom 𝐺, (𝐹 supp ∅)))
120119ad2antrr 724 . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . 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 7073 . . . . . . . . . . . . . . . 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 834 . . . . . . . . . . . . . . 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 7520 . . . . . . . . . . . . . . . 16 suc 𝑦 ∈ V
125124epeli 5462 . . . . . . . . . . . . . . 15 (𝑦 E suc 𝑦𝑦 ∈ suc 𝑦)
126 fvex 6677 . . . . . . . . . . . . . . . 16 (𝐺‘suc 𝑦) ∈ V
127126epeli 5462 . . . . . . . . . . . . . . 15 ((𝐺𝑦) E (𝐺‘suc 𝑦) ↔ (𝐺𝑦) ∈ (𝐺‘suc 𝑦))
128123, 125, 1273bitr3g 315 . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . 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 7527 . . . . . . . . . . . . 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 6844 . . . . . . . . . . . . . . . . 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 3967 . . . . . . . . . . . . . . 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 3967 . . . . . . . . . . . . . 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 7522 . . . . . . . . . . . . . 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 8211 . . . . . . . . . . . . 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 1369 . . . . . . . . . . . 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 724 . . . . . . . . . . . . . 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 6846 . . . . . . . . . . . . 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 7580 . . . . . . . . . . . . . . 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 8156 . . . . . . . . . . . . . . 15 ((ω ∈ On ∧ (𝐺𝑦) ∈ On) → (ω ↑o (𝐺𝑦)) ∈ On)
148138, 135, 147syl2anc 586 . . . . . . . . . . . . . 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 8206 . . . . . . . . . . . . . . 15 (((ω ∈ On ∧ (𝐺𝑦) ∈ On) ∧ ∅ ∈ ω) → ∅ ∈ (ω ↑o (𝐺𝑦)))
150138, 135, 139, 149syl21anc 835 . . . . . . . . . . . . . 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 8187 . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . 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 234 . . . . . . . . . . . 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 8146 . . . . . . . . . . . . 13 ((ω ∈ On ∧ (𝐺𝑦) ∈ On) → (ω ↑o suc (𝐺𝑦)) = ((ω ↑o (𝐺𝑦)) ·o ω))
155138, 135, 154syl2anc 586 . . . . . . . . . . . 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 2916 . . . . . . . . . . 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 3967 . . . . . . . . . 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 771 . . . . . . . . . 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 9156 . . . . . . . . 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 423 . . . . . . . 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 416 . . . . 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 7604 . . . 4 (𝑤 ∈ ω → (𝜑 → (𝑤 ∈ dom 𝐺 → (𝑇‘suc 𝑤):(𝐻‘suc 𝑤)–1-1-onto→((ω ↑o (𝐺𝑤)) ·o (𝐹‘(𝐺𝑤))))))
16529, 164vtoclga 3573 . . 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 208  wa 398   = wceq 1533  wcel 2110  Vcvv 3494  cun 3933  wss 3935  c0 4290   class class class wbr 5058  cmpt 5138  Tr wtr 5164   E cep 5458   We wwe 5507  ccnv 5548  dom cdm 5549  Ord word 6184  Oncon0 6185  suc csuc 6187  wf 6345  1-1-ontowf1o 6348  cfv 6349   Isom wiso 6350  (class class class)co 7150  cmpo 7152  ωcom 7574   supp csupp 7824  seqωcseqom 8077   +o coa 8093   ·o comu 8094  o coe 8095   finSupp cfsupp 8827  OrdIsocoi 8967   CNF ccnf 9118
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-inf2 9098
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-supp 7825  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-seqom 8078  df-1o 8096  df-2o 8097  df-oadd 8100  df-omul 8101  df-oexp 8102  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fsupp 8828  df-oi 8968  df-cnf 9119
This theorem is referenced by:  cnfcom2  9159
  Copyright terms: Public domain W3C validator