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Theorem cantnfvalf 8725
Description: Lemma for cantnf 8753. The function appearing in cantnfval 8728 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 7702 . 2 𝐹 Fn ω
3 nn0suc 7236 . . . 4 (𝑥 ∈ ω → (𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦))
4 fveq2 6332 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
5 0ex 4924 . . . . . . . 8 ∅ ∈ V
61seqom0g 7703 . . . . . . . 8 (∅ ∈ V → (𝐹‘∅) = ∅)
75, 6ax-mp 5 . . . . . . 7 (𝐹‘∅) = ∅
84, 7syl6eq 2821 . . . . . 6 (𝑥 = ∅ → (𝐹𝑥) = ∅)
9 0elon 5921 . . . . . 6 ∅ ∈ On
108, 9syl6eqel 2858 . . . . 5 (𝑥 = ∅ → (𝐹𝑥) ∈ On)
111seqomsuc 7704 . . . . . . . . 9 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = (𝑦(𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))(𝐹𝑦)))
12 df-ov 6795 . . . . . . . . 9 (𝑦(𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))(𝐹𝑦)) = ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))‘⟨𝑦, (𝐹𝑦)⟩)
1311, 12syl6eq 2821 . . . . . . . 8 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))‘⟨𝑦, (𝐹𝑦)⟩))
14 df-ov 6795 . . . . . . . . . . . 12 (𝐶 +𝑜 𝐷) = ( +𝑜 ‘⟨𝐶, 𝐷⟩)
15 fnoa 7741 . . . . . . . . . . . . . 14 +𝑜 Fn (On × On)
16 oacl 7768 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 +𝑜 𝑦) ∈ On)
1716rgen2a 3126 . . . . . . . . . . . . . 14 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +𝑜 𝑦) ∈ On
18 ffnov 6910 . . . . . . . . . . . . . 14 ( +𝑜 :(On × On)⟶On ↔ ( +𝑜 Fn (On × On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +𝑜 𝑦) ∈ On))
1915, 17, 18mpbir2an 682 . . . . . . . . . . . . 13 +𝑜 :(On × On)⟶On
2019, 9f0cli 6513 . . . . . . . . . . . 12 ( +𝑜 ‘⟨𝐶, 𝐷⟩) ∈ On
2114, 20eqeltri 2846 . . . . . . . . . . 11 (𝐶 +𝑜 𝐷) ∈ On
2221rgen2w 3074 . . . . . . . . . 10 𝑘𝐴𝑧𝐵 (𝐶 +𝑜 𝐷) ∈ On
23 eqid 2771 . . . . . . . . . . 11 (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷)) = (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))
2423fmpt2 7386 . . . . . . . . . 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 6513 . . . . . . . 8 ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +𝑜 𝐷))‘⟨𝑦, (𝐹𝑦)⟩) ∈ On
2713, 26syl6eqel 2858 . . . . . . 7 (𝑦 ∈ ω → (𝐹‘suc 𝑦) ∈ On)
28 fveq2 6332 . . . . . . . 8 (𝑥 = suc 𝑦 → (𝐹𝑥) = (𝐹‘suc 𝑦))
2928eleq1d 2835 . . . . . . 7 (𝑥 = suc 𝑦 → ((𝐹𝑥) ∈ On ↔ (𝐹‘suc 𝑦) ∈ On))
3027, 29syl5ibrcom 237 . . . . . 6 (𝑦 ∈ ω → (𝑥 = suc 𝑦 → (𝐹𝑥) ∈ On))
3130rexlimiv 3175 . . . . 5 (∃𝑦 ∈ ω 𝑥 = suc 𝑦 → (𝐹𝑥) ∈ On)
3210, 31jaoi 836 . . . 4 ((𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦) → (𝐹𝑥) ∈ On)
333, 32syl 17 . . 3 (𝑥 ∈ ω → (𝐹𝑥) ∈ On)
3433rgen 3071 . 2 𝑥 ∈ ω (𝐹𝑥) ∈ On
35 ffnfv 6530 . 2 (𝐹:ω⟶On ↔ (𝐹 Fn ω ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ On))
362, 34, 35mpbir2an 682 1 𝐹:ω⟶On
Colors of variables: wff setvar class
Syntax hints:  wo 826   = wceq 1631  wcel 2145  wral 3061  wrex 3062  Vcvv 3351  c0 4063  cop 4322   × cxp 5247  Oncon0 5866  suc csuc 5868   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6792  cmpt2 6794  ωcom 7211  seq𝜔cseqom 7694   +𝑜 coa 7709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-seqom 7695  df-oadd 7716
This theorem is referenced by:  cantnfval2  8729  cantnfle  8731  cantnflt  8732  cantnflem1d  8748  cantnflem1  8749  cnfcomlem  8759
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