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Theorem findreccl 36436
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 8466 . . 3 (𝐴𝑃 → (rec(𝐺, 𝐴)‘∅) = 𝐴)
2 eleq1a 2834 . . 3 (𝐴𝑃 → ((rec(𝐺, 𝐴)‘∅) = 𝐴 → (rec(𝐺, 𝐴)‘∅) ∈ 𝑃))
31, 2mpd 15 . 2 (𝐴𝑃 → (rec(𝐺, 𝐴)‘∅) ∈ 𝑃)
4 nnon 7893 . . . 4 (𝑦 ∈ ω → 𝑦 ∈ On)
5 fveq2 6907 . . . . . . 7 (𝑧 = (rec(𝐺, 𝐴)‘𝑦) → (𝐺𝑧) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
65eleq1d 2824 . . . . . 6 (𝑧 = (rec(𝐺, 𝐴)‘𝑦) → ((𝐺𝑧) ∈ 𝑃 ↔ (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃))
7 findreccl.1 . . . . . 6 (𝑧𝑃 → (𝐺𝑧) ∈ 𝑃)
86, 7vtoclga 3577 . . . . 5 ((rec(𝐺, 𝐴)‘𝑦) ∈ 𝑃 → (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃)
9 rdgsuc 8463 . . . . . 6 (𝑦 ∈ On → (rec(𝐺, 𝐴)‘suc 𝑦) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
109eleq1d 2824 . . . . 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 36435 1 (𝐶 ∈ ω → (𝐴𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  c0 4339  Oncon0 6386  suc csuc 6388  cfv 6563  ωcom 7887  reccrdg 8448
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-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449
This theorem is referenced by:  findabrcl  36437
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