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Theorem findreccl 36455
Description: Please add description here. (Contributed by Jeff Hoffman, 19-Feb-2008.)
Hypothesis
Ref Expression
findreccl.1 (𝑧𝑃 → (𝐺𝑧) ∈ 𝑃)
Assertion
Ref Expression
findreccl (𝐶 ∈ ω → (𝐴𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))
Distinct variable groups:   𝑧,𝐺   𝑧,𝐴   𝑧,𝑃
Allowed substitution hint:   𝐶(𝑧)

Proof of Theorem findreccl
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 rdg0g 8468 . . 3 (𝐴𝑃 → (rec(𝐺, 𝐴)‘∅) = 𝐴)
2 eleq1a 2835 . . 3 (𝐴𝑃 → ((rec(𝐺, 𝐴)‘∅) = 𝐴 → (rec(𝐺, 𝐴)‘∅) ∈ 𝑃))
31, 2mpd 15 . 2 (𝐴𝑃 → (rec(𝐺, 𝐴)‘∅) ∈ 𝑃)
4 nnon 7894 . . . 4 (𝑦 ∈ ω → 𝑦 ∈ On)
5 fveq2 6905 . . . . . . 7 (𝑧 = (rec(𝐺, 𝐴)‘𝑦) → (𝐺𝑧) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
65eleq1d 2825 . . . . . 6 (𝑧 = (rec(𝐺, 𝐴)‘𝑦) → ((𝐺𝑧) ∈ 𝑃 ↔ (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃))
7 findreccl.1 . . . . . 6 (𝑧𝑃 → (𝐺𝑧) ∈ 𝑃)
86, 7vtoclga 3576 . . . . 5 ((rec(𝐺, 𝐴)‘𝑦) ∈ 𝑃 → (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃)
9 rdgsuc 8465 . . . . . 6 (𝑦 ∈ On → (rec(𝐺, 𝐴)‘suc 𝑦) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
109eleq1d 2825 . . . . 5 (𝑦 ∈ On → ((rec(𝐺, 𝐴)‘suc 𝑦) ∈ 𝑃 ↔ (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃))
118, 10imbitrrid 246 . . . 4 (𝑦 ∈ On → ((rec(𝐺, 𝐴)‘𝑦) ∈ 𝑃 → (rec(𝐺, 𝐴)‘suc 𝑦) ∈ 𝑃))
124, 11syl 17 . . 3 (𝑦 ∈ ω → ((rec(𝐺, 𝐴)‘𝑦) ∈ 𝑃 → (rec(𝐺, 𝐴)‘suc 𝑦) ∈ 𝑃))
1312a1d 25 . 2 (𝑦 ∈ ω → (𝐴𝑃 → ((rec(𝐺, 𝐴)‘𝑦) ∈ 𝑃 → (rec(𝐺, 𝐴)‘suc 𝑦) ∈ 𝑃)))
143, 13findfvcl 36454 1 (𝐶 ∈ ω → (𝐴𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  c0 4332  Oncon0 6383  suc csuc 6385  cfv 6560  ωcom 7888  reccrdg 8450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451
This theorem is referenced by:  findabrcl  36456
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