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Theorem cantnfvalf 9655
Description: Lemma for cantnf 9683. The function appearing in cantnfval 9658 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 8450 . 2 𝐹 Fn ω
3 nn0suc 7879 . . . 4 (𝑥 ∈ ω → (𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦))
4 fveq2 6881 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
5 0ex 5297 . . . . . . . 8 ∅ ∈ V
61seqom0g 8451 . . . . . . . 8 (∅ ∈ V → (𝐹‘∅) = ∅)
75, 6ax-mp 5 . . . . . . 7 (𝐹‘∅) = ∅
84, 7eqtrdi 2780 . . . . . 6 (𝑥 = ∅ → (𝐹𝑥) = ∅)
9 0elon 6408 . . . . . 6 ∅ ∈ On
108, 9eqeltrdi 2833 . . . . 5 (𝑥 = ∅ → (𝐹𝑥) ∈ On)
111seqomsuc 8452 . . . . . . . . 9 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = (𝑦(𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))(𝐹𝑦)))
12 df-ov 7404 . . . . . . . . 9 (𝑦(𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))(𝐹𝑦)) = ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹𝑦)⟩)
1311, 12eqtrdi 2780 . . . . . . . 8 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹𝑦)⟩))
14 df-ov 7404 . . . . . . . . . . . 12 (𝐶 +o 𝐷) = ( +o ‘⟨𝐶, 𝐷⟩)
15 fnoa 8503 . . . . . . . . . . . . . 14 +o Fn (On × On)
16 oacl 8530 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 +o 𝑦) ∈ On)
1716rgen2 3189 . . . . . . . . . . . . . 14 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On
18 ffnov 7527 . . . . . . . . . . . . . 14 ( +o :(On × On)⟶On ↔ ( +o Fn (On × On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On))
1915, 17, 18mpbir2an 708 . . . . . . . . . . . . 13 +o :(On × On)⟶On
2019, 9f0cli 7089 . . . . . . . . . . . 12 ( +o ‘⟨𝐶, 𝐷⟩) ∈ On
2114, 20eqeltri 2821 . . . . . . . . . . 11 (𝐶 +o 𝐷) ∈ On
2221rgen2w 3058 . . . . . . . . . 10 𝑘𝐴𝑧𝐵 (𝐶 +o 𝐷) ∈ On
23 eqid 2724 . . . . . . . . . . 11 (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷)) = (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))
2423fmpo 8047 . . . . . . . . . 10 (∀𝑘𝐴𝑧𝐵 (𝐶 +o 𝐷) ∈ On ↔ (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On)
2522, 24mpbi 229 . . . . . . . . 9 (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On
2625, 9f0cli 7089 . . . . . . . 8 ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹𝑦)⟩) ∈ On
2713, 26eqeltrdi 2833 . . . . . . 7 (𝑦 ∈ ω → (𝐹‘suc 𝑦) ∈ On)
28 fveq2 6881 . . . . . . . 8 (𝑥 = suc 𝑦 → (𝐹𝑥) = (𝐹‘suc 𝑦))
2928eleq1d 2810 . . . . . . 7 (𝑥 = suc 𝑦 → ((𝐹𝑥) ∈ On ↔ (𝐹‘suc 𝑦) ∈ On))
3027, 29syl5ibrcom 246 . . . . . 6 (𝑦 ∈ ω → (𝑥 = suc 𝑦 → (𝐹𝑥) ∈ On))
3130rexlimiv 3140 . . . . 5 (∃𝑦 ∈ ω 𝑥 = suc 𝑦 → (𝐹𝑥) ∈ On)
3210, 31jaoi 854 . . . 4 ((𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦) → (𝐹𝑥) ∈ On)
333, 32syl 17 . . 3 (𝑥 ∈ ω → (𝐹𝑥) ∈ On)
3433rgen 3055 . 2 𝑥 ∈ ω (𝐹𝑥) ∈ On
35 ffnfv 7110 . 2 (𝐹:ω⟶On ↔ (𝐹 Fn ω ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ On))
362, 34, 35mpbir2an 708 1 𝐹:ω⟶On
Colors of variables: wff setvar class
Syntax hints:  wo 844   = wceq 1533  wcel 2098  wral 3053  wrex 3062  Vcvv 3466  c0 4314  cop 4626   × cxp 5664  Oncon0 6354  suc csuc 6356   Fn wfn 6528  wf 6529  cfv 6533  (class class class)co 7401  cmpo 7403  ωcom 7848  seqωcseqom 8442   +o coa 8458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-seqom 8443  df-oadd 8465
This theorem is referenced by:  cantnfval2  9659  cantnfle  9661  cantnflt  9662  cantnflem1d  9678  cantnflem1  9679  cnfcomlem  9689
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