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Theorem cantnfvalf 8963
Description: Lemma for cantnf 8991. The function appearing in cantnfval 8966 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 7933 . 2 𝐹 Fn ω
3 nn0suc 7453 . . . 4 (𝑥 ∈ ω → (𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦))
4 fveq2 6530 . . . . . . 7 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
5 0ex 5096 . . . . . . . 8 ∅ ∈ V
61seqom0g 7934 . . . . . . . 8 (∅ ∈ V → (𝐹‘∅) = ∅)
75, 6ax-mp 5 . . . . . . 7 (𝐹‘∅) = ∅
84, 7syl6eq 2845 . . . . . 6 (𝑥 = ∅ → (𝐹𝑥) = ∅)
9 0elon 6111 . . . . . 6 ∅ ∈ On
108, 9syl6eqel 2889 . . . . 5 (𝑥 = ∅ → (𝐹𝑥) ∈ On)
111seqomsuc 7935 . . . . . . . . 9 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = (𝑦(𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))(𝐹𝑦)))
12 df-ov 7010 . . . . . . . . 9 (𝑦(𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))(𝐹𝑦)) = ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹𝑦)⟩)
1311, 12syl6eq 2845 . . . . . . . 8 (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹𝑦)⟩))
14 df-ov 7010 . . . . . . . . . . . 12 (𝐶 +o 𝐷) = ( +o ‘⟨𝐶, 𝐷⟩)
15 fnoa 7975 . . . . . . . . . . . . . 14 +o Fn (On × On)
16 oacl 8002 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 +o 𝑦) ∈ On)
1716rgen2a 3191 . . . . . . . . . . . . . 14 𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On
18 ffnov 7125 . . . . . . . . . . . . . 14 ( +o :(On × On)⟶On ↔ ( +o Fn (On × On) ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On))
1915, 17, 18mpbir2an 707 . . . . . . . . . . . . 13 +o :(On × On)⟶On
2019, 9f0cli 6718 . . . . . . . . . . . 12 ( +o ‘⟨𝐶, 𝐷⟩) ∈ On
2114, 20eqeltri 2877 . . . . . . . . . . 11 (𝐶 +o 𝐷) ∈ On
2221rgen2w 3116 . . . . . . . . . 10 𝑘𝐴𝑧𝐵 (𝐶 +o 𝐷) ∈ On
23 eqid 2793 . . . . . . . . . . 11 (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷)) = (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))
2423fmpo 7613 . . . . . . . . . 10 (∀𝑘𝐴𝑧𝐵 (𝐶 +o 𝐷) ∈ On ↔ (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On)
2522, 24mpbi 231 . . . . . . . . 9 (𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On
2625, 9f0cli 6718 . . . . . . . 8 ((𝑘𝐴, 𝑧𝐵 ↦ (𝐶 +o 𝐷))‘⟨𝑦, (𝐹𝑦)⟩) ∈ On
2713, 26syl6eqel 2889 . . . . . . 7 (𝑦 ∈ ω → (𝐹‘suc 𝑦) ∈ On)
28 fveq2 6530 . . . . . . . 8 (𝑥 = suc 𝑦 → (𝐹𝑥) = (𝐹‘suc 𝑦))
2928eleq1d 2865 . . . . . . 7 (𝑥 = suc 𝑦 → ((𝐹𝑥) ∈ On ↔ (𝐹‘suc 𝑦) ∈ On))
3027, 29syl5ibrcom 248 . . . . . 6 (𝑦 ∈ ω → (𝑥 = suc 𝑦 → (𝐹𝑥) ∈ On))
3130rexlimiv 3240 . . . . 5 (∃𝑦 ∈ ω 𝑥 = suc 𝑦 → (𝐹𝑥) ∈ On)
3210, 31jaoi 852 . . . 4 ((𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦) → (𝐹𝑥) ∈ On)
333, 32syl 17 . . 3 (𝑥 ∈ ω → (𝐹𝑥) ∈ On)
3433rgen 3113 . 2 𝑥 ∈ ω (𝐹𝑥) ∈ On
35 ffnfv 6736 . 2 (𝐹:ω⟶On ↔ (𝐹 Fn ω ∧ ∀𝑥 ∈ ω (𝐹𝑥) ∈ On))
362, 34, 35mpbir2an 707 1 𝐹:ω⟶On
Colors of variables: wff setvar class
Syntax hints:  wo 842   = wceq 1520  wcel 2079  wral 3103  wrex 3104  Vcvv 3432  c0 4206  cop 4472   × cxp 5433  Oncon0 6058  suc csuc 6060   Fn wfn 6212  wf 6213  cfv 6217  (class class class)co 7007  cmpo 7009  ωcom 7427  seq𝜔cseqom 7925   +o coa 7941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-seqom 7926  df-oadd 7948
This theorem is referenced by:  cantnfval2  8967  cantnfle  8969  cantnflt  8970  cantnflem1d  8986  cantnflem1  8987  cnfcomlem  8997
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