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Theorem findreccl 33828
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 8055 . . 3 (𝐴𝑃 → (rec(𝐺, 𝐴)‘∅) = 𝐴)
2 eleq1a 2911 . . 3 (𝐴𝑃 → ((rec(𝐺, 𝐴)‘∅) = 𝐴 → (rec(𝐺, 𝐴)‘∅) ∈ 𝑃))
31, 2mpd 15 . 2 (𝐴𝑃 → (rec(𝐺, 𝐴)‘∅) ∈ 𝑃)
4 nnon 7577 . . . 4 (𝑦 ∈ ω → 𝑦 ∈ On)
5 fveq2 6659 . . . . . . 7 (𝑧 = (rec(𝐺, 𝐴)‘𝑦) → (𝐺𝑧) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
65eleq1d 2900 . . . . . 6 (𝑧 = (rec(𝐺, 𝐴)‘𝑦) → ((𝐺𝑧) ∈ 𝑃 ↔ (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃))
7 findreccl.1 . . . . . 6 (𝑧𝑃 → (𝐺𝑧) ∈ 𝑃)
86, 7vtoclga 3560 . . . . 5 ((rec(𝐺, 𝐴)‘𝑦) ∈ 𝑃 → (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃)
9 rdgsuc 8052 . . . . . 6 (𝑦 ∈ On → (rec(𝐺, 𝐴)‘suc 𝑦) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
109eleq1d 2900 . . . . 5 (𝑦 ∈ On → ((rec(𝐺, 𝐴)‘suc 𝑦) ∈ 𝑃 ↔ (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃))
118, 10syl5ibr 249 . . . 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 33827 1 (𝐶 ∈ ω → (𝐴𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  c0 4276  Oncon0 6179  suc csuc 6181  cfv 6344  ωcom 7571  reccrdg 8037
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-om 7572  df-wrecs 7939  df-recs 8000  df-rdg 8038
This theorem is referenced by:  findabrcl  33829
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