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Theorem cantnfvalf 9280
Description: Lemma for cantnf 9308. The function appearing in cantnfval 9283 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 8191 . 2 𝐹 Fn ω
3 nn0suc 7673 . . . 4 (𝑥 ∈ ω → (𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦))
4 fveq2 6717 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
5 0ex 5200 . . . . . . . 8 ∅ ∈ V
61seqom0g 8192 . . . . . . . 8 (∅ ∈ V → (𝐹‘∅) = ∅)
75, 6ax-mp 5 . . . . . . 7 (𝐹‘∅) = ∅
84, 7eqtrdi 2794 . . . . . 6 (𝑥 = ∅ → (𝐹𝑥) = ∅)
9 0elon 6266 . . . . . 6 ∅ ∈ On
108, 9eqeltrdi 2846 . . . . 5 (𝑥 = ∅ → (𝐹𝑥) ∈ On)
111seqomsuc 8193 . . . . . . . . 9 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = (𝑦(𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))(𝐹𝑦)))
12 df-ov 7216 . . . . . . . . 9 (𝑦(𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))(𝐹𝑦)) = ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹𝑦)⟩)
1311, 12eqtrdi 2794 . . . . . . . 8 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹𝑦)⟩))
14 df-ov 7216 . . . . . . . . . . . 12 (𝐶 +o 𝐷) = ( +o ‘⟨𝐶, 𝐷⟩)
15 fnoa 8235 . . . . . . . . . . . . . 14 +o Fn (On × On)
16 oacl 8262 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 +o 𝑦) ∈ On)
1716rgen2 3124 . . . . . . . . . . . . . 14 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On
18 ffnov 7337 . . . . . . . . . . . . . 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 6917 . . . . . . . . . . . 12 ( +o ‘⟨𝐶, 𝐷⟩) ∈ On
2114, 20eqeltri 2834 . . . . . . . . . . 11 (𝐶 +o 𝐷) ∈ On
2221rgen2w 3074 . . . . . . . . . 10 𝑘𝐴𝑧𝐵 (𝐶 +o 𝐷) ∈ On
23 eqid 2737 . . . . . . . . . . 11 (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷)) = (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))
2423fmpo 7838 . . . . . . . . . 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 6917 . . . . . . . 8 ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹𝑦)⟩) ∈ On
2713, 26eqeltrdi 2846 . . . . . . 7 (𝑦 ∈ ω → (𝐹‘suc 𝑦) ∈ On)
28 fveq2 6717 . . . . . . . 8 (𝑥 = suc 𝑦 → (𝐹𝑥) = (𝐹‘suc 𝑦))
2928eleq1d 2822 . . . . . . 7 (𝑥 = suc 𝑦 → ((𝐹𝑥) ∈ On ↔ (𝐹‘suc 𝑦) ∈ On))
3027, 29syl5ibrcom 250 . . . . . 6 (𝑦 ∈ ω → (𝑥 = suc 𝑦 → (𝐹𝑥) ∈ On))
3130rexlimiv 3199 . . . . 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 3071 . 2 𝑥 ∈ ω (𝐹𝑥) ∈ On
35 ffnfv 6935 . 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 1543  wcel 2110  wral 3061  wrex 3062  Vcvv 3408  c0 4237  cop 4547   × cxp 5549  Oncon0 6213  suc csuc 6215   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  cmpo 7215  ωcom 7644  seqωcseqom 8183   +o coa 8199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-seqom 8184  df-oadd 8206
This theorem is referenced by:  cantnfval2  9284  cantnfle  9286  cantnflt  9287  cantnflem1d  9303  cantnflem1  9304  cnfcomlem  9314
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