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Theorem findreccl 32679
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 7643 . . 3 (𝐴𝑃 → (rec(𝐺, 𝐴)‘∅) = 𝐴)
2 eleq1a 2798 . . 3 (𝐴𝑃 → ((rec(𝐺, 𝐴)‘∅) = 𝐴 → (rec(𝐺, 𝐴)‘∅) ∈ 𝑃))
31, 2mpd 15 . 2 (𝐴𝑃 → (rec(𝐺, 𝐴)‘∅) ∈ 𝑃)
4 nnon 7188 . . . 4 (𝑦 ∈ ω → 𝑦 ∈ On)
5 fveq2 6304 . . . . . . 7 (𝑧 = (rec(𝐺, 𝐴)‘𝑦) → (𝐺𝑧) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
65eleq1d 2788 . . . . . 6 (𝑧 = (rec(𝐺, 𝐴)‘𝑦) → ((𝐺𝑧) ∈ 𝑃 ↔ (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃))
7 findreccl.1 . . . . . 6 (𝑧𝑃 → (𝐺𝑧) ∈ 𝑃)
86, 7vtoclga 3376 . . . . 5 ((rec(𝐺, 𝐴)‘𝑦) ∈ 𝑃 → (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃)
9 rdgsuc 7640 . . . . . 6 (𝑦 ∈ On → (rec(𝐺, 𝐴)‘suc 𝑦) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
109eleq1d 2788 . . . . 5 (𝑦 ∈ On → ((rec(𝐺, 𝐴)‘suc 𝑦) ∈ 𝑃 ↔ (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃))
118, 10syl5ibr 236 . . . 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 32678 1 (𝐶 ∈ ω → (𝐴𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1596   ∈ wcel 2103  ∅c0 4023  Oncon0 5836  suc csuc 5838  ‘cfv 6001  ωcom 7182  reccrdg 7625 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-om 7183  df-wrecs 7527  df-recs 7588  df-rdg 7626 This theorem is referenced by:  findabrcl  32680
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