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Theorem cantnfvalf 8507
 Description: Lemma for cantnf 8535. The function appearing in cantnfval 8510 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.)
Hypothesis
Ref Expression
cantnfvalf.f 𝐹 = seq𝜔((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷)), ∅)
Assertion
Ref Expression
cantnfvalf 𝐹:ω⟶On
Distinct variable groups:   𝑧,𝑘,𝐴   𝐵,𝑘,𝑧
Allowed substitution hints:   𝐶(𝑧,𝑘)   𝐷(𝑧,𝑘)   𝐹(𝑧,𝑘)

Proof of Theorem cantnfvalf
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfvalf.f . . 3 𝐹 = seq𝜔((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷)), ∅)
21fnseqom 7496 . 2 𝐹 Fn ω
3 nn0suc 7038 . . . 4 (𝑥 ∈ ω → (𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦))
4 fveq2 6150 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
5 0ex 4755 . . . . . . . 8 ∅ ∈ V
61seqom0g 7497 . . . . . . . 8 (∅ ∈ V → (𝐹‘∅) = ∅)
75, 6ax-mp 5 . . . . . . 7 (𝐹‘∅) = ∅
84, 7syl6eq 2676 . . . . . 6 (𝑥 = ∅ → (𝐹𝑥) = ∅)
9 0elon 5740 . . . . . 6 ∅ ∈ On
108, 9syl6eqel 2712 . . . . 5 (𝑥 = ∅ → (𝐹𝑥) ∈ On)
111seqomsuc 7498 . . . . . . . . 9 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = (𝑦(𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))(𝐹𝑦)))
12 df-ov 6608 . . . . . . . . 9 (𝑦(𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))(𝐹𝑦)) = ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))‘⟨𝑦, (𝐹𝑦)⟩)
1311, 12syl6eq 2676 . . . . . . . 8 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))‘⟨𝑦, (𝐹𝑦)⟩))
14 df-ov 6608 . . . . . . . . . . . 12 (𝐶 +𝑜 𝐷) = ( +𝑜 ‘⟨𝐶, 𝐷⟩)
15 fnoa 7534 . . . . . . . . . . . . . 14 +𝑜 Fn (On × On)
16 oacl 7561 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 +𝑜 𝑦) ∈ On)
1716rgen2a 2976 . . . . . . . . . . . . . 14 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +𝑜 𝑦) ∈ On
18 ffnov 6718 . . . . . . . . . . . . . 14 ( +𝑜 :(On × On)⟶On ↔ ( +𝑜 Fn (On × On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +𝑜 𝑦) ∈ On))
1915, 17, 18mpbir2an 954 . . . . . . . . . . . . 13 +𝑜 :(On × On)⟶On
2019, 9f0cli 6327 . . . . . . . . . . . 12 ( +𝑜 ‘⟨𝐶, 𝐷⟩) ∈ On
2114, 20eqeltri 2700 . . . . . . . . . . 11 (𝐶 +𝑜 𝐷) ∈ On
2221rgen2w 2925 . . . . . . . . . 10 𝑘𝐴𝑧𝐵 (𝐶 +𝑜 𝐷) ∈ On
23 eqid 2626 . . . . . . . . . . 11 (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷)) = (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))
2423fmpt2 7183 . . . . . . . . . 10 (∀𝑘𝐴𝑧𝐵 (𝐶 +𝑜 𝐷) ∈ On ↔ (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷)):(𝐴 × 𝐵)⟶On)
2522, 24mpbi 220 . . . . . . . . 9 (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷)):(𝐴 × 𝐵)⟶On
2625, 9f0cli 6327 . . . . . . . 8 ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))‘⟨𝑦, (𝐹𝑦)⟩) ∈ On
2713, 26syl6eqel 2712 . . . . . . 7 (𝑦 ∈ ω → (𝐹‘suc 𝑦) ∈ On)
28 fveq2 6150 . . . . . . . 8 (𝑥 = suc 𝑦 → (𝐹𝑥) = (𝐹‘suc 𝑦))
2928eleq1d 2688 . . . . . . 7 (𝑥 = suc 𝑦 → ((𝐹𝑥) ∈ On ↔ (𝐹‘suc 𝑦) ∈ On))
3027, 29syl5ibrcom 237 . . . . . 6 (𝑦 ∈ ω → (𝑥 = suc 𝑦 → (𝐹𝑥) ∈ On))
3130rexlimiv 3025 . . . . 5 (∃𝑦 ∈ ω 𝑥 = suc 𝑦 → (𝐹𝑥) ∈ On)
3210, 31jaoi 394 . . . 4 ((𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦) → (𝐹𝑥) ∈ On)
333, 32syl 17 . . 3 (𝑥 ∈ ω → (𝐹𝑥) ∈ On)
3433rgen 2922 . 2 𝑥 ∈ ω (𝐹𝑥) ∈ On
35 ffnfv 6344 . 2 (𝐹:ω⟶On ↔ (𝐹 Fn ω ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ On))
362, 34, 35mpbir2an 954 1 𝐹:ω⟶On
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 383   = wceq 1480   ∈ wcel 1992  ∀wral 2912  ∃wrex 2913  Vcvv 3191  ∅c0 3896  ⟨cop 4159   × cxp 5077  Oncon0 5685  suc csuc 5687   Fn wfn 5845  ⟶wf 5846  ‘cfv 5850  (class class class)co 6605   ↦ cmpt2 6607  ωcom 7013  seq𝜔cseqom 7488   +𝑜 coa 7503 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-seqom 7489  df-oadd 7510 This theorem is referenced by:  cantnfval2  8511  cantnfle  8513  cantnflt  8514  cantnflem1d  8530  cantnflem1  8531  cnfcomlem  8541
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