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Theorem cantnfvalf 8735
Description: Lemma for cantnf 8763. The function appearing in cantnfval 8738 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 7719 . 2 𝐹 Fn ω
3 nn0suc 7255 . . . 4 (𝑥 ∈ ω → (𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦))
4 fveq2 6352 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
5 0ex 4942 . . . . . . . 8 ∅ ∈ V
61seqom0g 7720 . . . . . . . 8 (∅ ∈ V → (𝐹‘∅) = ∅)
75, 6ax-mp 5 . . . . . . 7 (𝐹‘∅) = ∅
84, 7syl6eq 2810 . . . . . 6 (𝑥 = ∅ → (𝐹𝑥) = ∅)
9 0elon 5939 . . . . . 6 ∅ ∈ On
108, 9syl6eqel 2847 . . . . 5 (𝑥 = ∅ → (𝐹𝑥) ∈ On)
111seqomsuc 7721 . . . . . . . . 9 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = (𝑦(𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))(𝐹𝑦)))
12 df-ov 6816 . . . . . . . . 9 (𝑦(𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))(𝐹𝑦)) = ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))‘⟨𝑦, (𝐹𝑦)⟩)
1311, 12syl6eq 2810 . . . . . . . 8 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))‘⟨𝑦, (𝐹𝑦)⟩))
14 df-ov 6816 . . . . . . . . . . . 12 (𝐶 +𝑜 𝐷) = ( +𝑜 ‘⟨𝐶, 𝐷⟩)
15 fnoa 7757 . . . . . . . . . . . . . 14 +𝑜 Fn (On × On)
16 oacl 7784 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 +𝑜 𝑦) ∈ On)
1716rgen2a 3115 . . . . . . . . . . . . . 14 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +𝑜 𝑦) ∈ On
18 ffnov 6929 . . . . . . . . . . . . . 14 ( +𝑜 :(On × On)⟶On ↔ ( +𝑜 Fn (On × On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +𝑜 𝑦) ∈ On))
1915, 17, 18mpbir2an 993 . . . . . . . . . . . . 13 +𝑜 :(On × On)⟶On
2019, 9f0cli 6533 . . . . . . . . . . . 12 ( +𝑜 ‘⟨𝐶, 𝐷⟩) ∈ On
2114, 20eqeltri 2835 . . . . . . . . . . 11 (𝐶 +𝑜 𝐷) ∈ On
2221rgen2w 3063 . . . . . . . . . 10 𝑘𝐴𝑧𝐵 (𝐶 +𝑜 𝐷) ∈ On
23 eqid 2760 . . . . . . . . . . 11 (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷)) = (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))
2423fmpt2 7405 . . . . . . . . . 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 6533 . . . . . . . 8 ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))‘⟨𝑦, (𝐹𝑦)⟩) ∈ On
2713, 26syl6eqel 2847 . . . . . . 7 (𝑦 ∈ ω → (𝐹‘suc 𝑦) ∈ On)
28 fveq2 6352 . . . . . . . 8 (𝑥 = suc 𝑦 → (𝐹𝑥) = (𝐹‘suc 𝑦))
2928eleq1d 2824 . . . . . . 7 (𝑥 = suc 𝑦 → ((𝐹𝑥) ∈ On ↔ (𝐹‘suc 𝑦) ∈ On))
3027, 29syl5ibrcom 237 . . . . . 6 (𝑦 ∈ ω → (𝑥 = suc 𝑦 → (𝐹𝑥) ∈ On))
3130rexlimiv 3165 . . . . 5 (∃𝑦 ∈ ω 𝑥 = suc 𝑦 → (𝐹𝑥) ∈ On)
3210, 31jaoi 393 . . . 4 ((𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦) → (𝐹𝑥) ∈ On)
333, 32syl 17 . . 3 (𝑥 ∈ ω → (𝐹𝑥) ∈ On)
3433rgen 3060 . 2 𝑥 ∈ ω (𝐹𝑥) ∈ On
35 ffnfv 6551 . 2 (𝐹:ω⟶On ↔ (𝐹 Fn ω ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ On))
362, 34, 35mpbir2an 993 1 𝐹:ω⟶On
Colors of variables: wff setvar class
Syntax hints:  wo 382   = wceq 1632  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  c0 4058  cop 4327   × cxp 5264  Oncon0 5884  suc csuc 5886   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  cmpt2 6815  ωcom 7230  seq𝜔cseqom 7711   +𝑜 coa 7726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-seqom 7712  df-oadd 7733
This theorem is referenced by:  cantnfval2  8739  cantnfle  8741  cantnflt  8742  cantnflem1d  8758  cantnflem1  8759  cnfcomlem  8769
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