Theorem findreccl 32452

Description: Please add description here. (Contributed by Jeff Hoffman, 19-Feb-2008.)

Hypothesis

Ref	Expression
findreccl.1	⊢ (𝑧 ∈ 𝑃 → (𝐺‘𝑧) ∈ 𝑃)

Assertion

Ref	Expression
findreccl	⊢ (𝐶 ∈ ω → (𝐴 ∈ 𝑃 → (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.

Step	Hyp	Ref	Expression
1		rdg0g 7523	. . 3 ⊢ (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘∅) = 𝐴)
2		eleq1a 2696	. . 3 ⊢ (𝐴 ∈ 𝑃 → ((rec(𝐺, 𝐴)‘∅) = 𝐴 → (rec(𝐺, 𝐴)‘∅) ∈ 𝑃))
3	1, 2	mpd 15	. 2 ⊢ (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘∅) ∈ 𝑃)
4		nnon 7071	. . . 4 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
5		fveq2 6191	. . . . . . 7 ⊢ (𝑧 = (rec(𝐺, 𝐴)‘𝑦) → (𝐺‘𝑧) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
6	5	eleq1d 2686	. . . . . 6 ⊢ (𝑧 = (rec(𝐺, 𝐴)‘𝑦) → ((𝐺‘𝑧) ∈ 𝑃 ↔ (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃))
7		findreccl.1	. . . . . 6 ⊢ (𝑧 ∈ 𝑃 → (𝐺‘𝑧) ∈ 𝑃)
8	6, 7	vtoclga 3272	. . . . 5 ⊢ ((rec(𝐺, 𝐴)‘𝑦) ∈ 𝑃 → (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃)
9		rdgsuc 7520	. . . . . 6 ⊢ (𝑦 ∈ On → (rec(𝐺, 𝐴)‘suc 𝑦) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
10	9	eleq1d 2686	. . . . 5 ⊢ (𝑦 ∈ On → ((rec(𝐺, 𝐴)‘suc 𝑦) ∈ 𝑃 ↔ (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃))
11	8, 10	syl5ibr 236	. . . 4 ⊢ (𝑦 ∈ On → ((rec(𝐺, 𝐴)‘𝑦) ∈ 𝑃 → (rec(𝐺, 𝐴)‘suc 𝑦) ∈ 𝑃))
12	4, 11	syl 17	. . 3 ⊢ (𝑦 ∈ ω → ((rec(𝐺, 𝐴)‘𝑦) ∈ 𝑃 → (rec(𝐺, 𝐴)‘suc 𝑦) ∈ 𝑃))
13	12	a1d 25	. 2 ⊢ (𝑦 ∈ ω → (𝐴 ∈ 𝑃 → ((rec(𝐺, 𝐴)‘𝑦) ∈ 𝑃 → (rec(𝐺, 𝐴)‘suc 𝑦) ∈ 𝑃)))
14	3, 13	findfvcl 32451	1 ⊢ (𝐶 ∈ ω → (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ∅c0 3915  Oncon0 5723  suc csuc 5725  ‘cfv 5888  ωcom 7065  reccrdg 7505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  findabrcl  32453