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Theorem cantnfvalf 9115
 Description: Lemma for cantnf 9143. The function appearing in cantnfval 9118 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 8077 . 2 𝐹 Fn ω
3 nn0suc 7589 . . . 4 (𝑥 ∈ ω → (𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦))
4 fveq2 6646 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
5 0ex 5176 . . . . . . . 8 ∅ ∈ V
61seqom0g 8078 . . . . . . . 8 (∅ ∈ V → (𝐹‘∅) = ∅)
75, 6ax-mp 5 . . . . . . 7 (𝐹‘∅) = ∅
84, 7eqtrdi 2849 . . . . . 6 (𝑥 = ∅ → (𝐹𝑥) = ∅)
9 0elon 6213 . . . . . 6 ∅ ∈ On
108, 9eqeltrdi 2898 . . . . 5 (𝑥 = ∅ → (𝐹𝑥) ∈ On)
111seqomsuc 8079 . . . . . . . . 9 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = (𝑦(𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))(𝐹𝑦)))
12 df-ov 7139 . . . . . . . . 9 (𝑦(𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))(𝐹𝑦)) = ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹𝑦)⟩)
1311, 12eqtrdi 2849 . . . . . . . 8 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹𝑦)⟩))
14 df-ov 7139 . . . . . . . . . . . 12 (𝐶 +o 𝐷) = ( +o ‘⟨𝐶, 𝐷⟩)
15 fnoa 8119 . . . . . . . . . . . . . 14 +o Fn (On × On)
16 oacl 8146 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 +o 𝑦) ∈ On)
1716rgen2 3168 . . . . . . . . . . . . . 14 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On
18 ffnov 7259 . . . . . . . . . . . . . 14 ( +o :(On × On)⟶On ↔ ( +o Fn (On × On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On))
1915, 17, 18mpbir2an 710 . . . . . . . . . . . . 13 +o :(On × On)⟶On
2019, 9f0cli 6842 . . . . . . . . . . . 12 ( +o ‘⟨𝐶, 𝐷⟩) ∈ On
2114, 20eqeltri 2886 . . . . . . . . . . 11 (𝐶 +o 𝐷) ∈ On
2221rgen2w 3119 . . . . . . . . . 10 𝑘𝐴𝑧𝐵 (𝐶 +o 𝐷) ∈ On
23 eqid 2798 . . . . . . . . . . 11 (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷)) = (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))
2423fmpo 7751 . . . . . . . . . 10 (∀𝑘𝐴𝑧𝐵 (𝐶 +o 𝐷) ∈ On ↔ (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On)
2522, 24mpbi 233 . . . . . . . . 9 (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On
2625, 9f0cli 6842 . . . . . . . 8 ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹𝑦)⟩) ∈ On
2713, 26eqeltrdi 2898 . . . . . . 7 (𝑦 ∈ ω → (𝐹‘suc 𝑦) ∈ On)
28 fveq2 6646 . . . . . . . 8 (𝑥 = suc 𝑦 → (𝐹𝑥) = (𝐹‘suc 𝑦))
2928eleq1d 2874 . . . . . . 7 (𝑥 = suc 𝑦 → ((𝐹𝑥) ∈ On ↔ (𝐹‘suc 𝑦) ∈ On))
3027, 29syl5ibrcom 250 . . . . . 6 (𝑦 ∈ ω → (𝑥 = suc 𝑦 → (𝐹𝑥) ∈ On))
3130rexlimiv 3239 . . . . 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 3116 . 2 𝑥 ∈ ω (𝐹𝑥) ∈ On
35 ffnfv 6860 . 2 (𝐹:ω⟶On ↔ (𝐹 Fn ω ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ On))
362, 34, 35mpbir2an 710 1 𝐹:ω⟶On
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 844   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  Vcvv 3441  ∅c0 4243  ⟨cop 4531   × cxp 5518  Oncon0 6160  suc csuc 6162   Fn wfn 6320  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136   ∈ cmpo 7138  ωcom 7563  seqωcseqom 8069   +o coa 8085 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-seqom 8070  df-oadd 8092 This theorem is referenced by:  cantnfval2  9119  cantnfle  9121  cantnflt  9122  cantnflem1d  9138  cantnflem1  9139  cnfcomlem  9149
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