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Theorem cantnfvalf 9703
Description: Lemma for cantnf 9731. The function appearing in cantnfval 9706 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 8494 . 2 𝐹 Fn ω
3 nn0suc 7917 . . . 4 (𝑥 ∈ ω → (𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦))
4 fveq2 6907 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
5 0ex 5313 . . . . . . . 8 ∅ ∈ V
61seqom0g 8495 . . . . . . . 8 (∅ ∈ V → (𝐹‘∅) = ∅)
75, 6ax-mp 5 . . . . . . 7 (𝐹‘∅) = ∅
84, 7eqtrdi 2791 . . . . . 6 (𝑥 = ∅ → (𝐹𝑥) = ∅)
9 0elon 6440 . . . . . 6 ∅ ∈ On
108, 9eqeltrdi 2847 . . . . 5 (𝑥 = ∅ → (𝐹𝑥) ∈ On)
111seqomsuc 8496 . . . . . . . . 9 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = (𝑦(𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))(𝐹𝑦)))
12 df-ov 7434 . . . . . . . . 9 (𝑦(𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))(𝐹𝑦)) = ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹𝑦)⟩)
1311, 12eqtrdi 2791 . . . . . . . 8 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹𝑦)⟩))
14 df-ov 7434 . . . . . . . . . . . 12 (𝐶 +o 𝐷) = ( +o ‘⟨𝐶, 𝐷⟩)
15 fnoa 8545 . . . . . . . . . . . . . 14 +o Fn (On × On)
16 oacl 8572 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 +o 𝑦) ∈ On)
1716rgen2 3197 . . . . . . . . . . . . . 14 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On
18 ffnov 7559 . . . . . . . . . . . . . 14 ( +o :(On × On)⟶On ↔ ( +o Fn (On × On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On))
1915, 17, 18mpbir2an 711 . . . . . . . . . . . . 13 +o :(On × On)⟶On
2019, 9f0cli 7118 . . . . . . . . . . . 12 ( +o ‘⟨𝐶, 𝐷⟩) ∈ On
2114, 20eqeltri 2835 . . . . . . . . . . 11 (𝐶 +o 𝐷) ∈ On
2221rgen2w 3064 . . . . . . . . . 10 𝑘𝐴𝑧𝐵 (𝐶 +o 𝐷) ∈ On
23 eqid 2735 . . . . . . . . . . 11 (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷)) = (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))
2423fmpo 8092 . . . . . . . . . 10 (∀𝑘𝐴𝑧𝐵 (𝐶 +o 𝐷) ∈ On ↔ (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On)
2522, 24mpbi 230 . . . . . . . . 9 (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On
2625, 9f0cli 7118 . . . . . . . 8 ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹𝑦)⟩) ∈ On
2713, 26eqeltrdi 2847 . . . . . . 7 (𝑦 ∈ ω → (𝐹‘suc 𝑦) ∈ On)
28 fveq2 6907 . . . . . . . 8 (𝑥 = suc 𝑦 → (𝐹𝑥) = (𝐹‘suc 𝑦))
2928eleq1d 2824 . . . . . . 7 (𝑥 = suc 𝑦 → ((𝐹𝑥) ∈ On ↔ (𝐹‘suc 𝑦) ∈ On))
3027, 29syl5ibrcom 247 . . . . . 6 (𝑦 ∈ ω → (𝑥 = suc 𝑦 → (𝐹𝑥) ∈ On))
3130rexlimiv 3146 . . . . 5 (∃𝑦 ∈ ω 𝑥 = suc 𝑦 → (𝐹𝑥) ∈ On)
3210, 31jaoi 857 . . . 4 ((𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦) → (𝐹𝑥) ∈ On)
333, 32syl 17 . . 3 (𝑥 ∈ ω → (𝐹𝑥) ∈ On)
3433rgen 3061 . 2 𝑥 ∈ ω (𝐹𝑥) ∈ On
35 ffnfv 7139 . 2 (𝐹:ω⟶On ↔ (𝐹 Fn ω ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ On))
362, 34, 35mpbir2an 711 1 𝐹:ω⟶On
Colors of variables: wff setvar class
Syntax hints:  wo 847   = wceq 1537  wcel 2106  wral 3059  wrex 3068  Vcvv 3478  c0 4339  cop 4637   × cxp 5687  Oncon0 6386  suc csuc 6388   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  ωcom 7887  seqωcseqom 8486   +o coa 8502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seqom 8487  df-oadd 8509
This theorem is referenced by:  cantnfval2  9707  cantnfle  9709  cantnflt  9710  cantnflem1d  9726  cantnflem1  9727  cnfcomlem  9737
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