Theorem findreccl 33801

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 8063	. . 3 ⊢ (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘∅) = 𝐴)
2		eleq1a 2908	. . 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 7586	. . . 4 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
5		fveq2 6670	. . . . . . 7 ⊢ (𝑧 = (rec(𝐺, 𝐴)‘𝑦) → (𝐺‘𝑧) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
6	5	eleq1d 2897	. . . . . 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 3574	. . . . 5 ⊢ ((rec(𝐺, 𝐴)‘𝑦) ∈ 𝑃 → (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃)
9		rdgsuc 8060	. . . . . 6 ⊢ (𝑦 ∈ On → (rec(𝐺, 𝐴)‘suc 𝑦) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
10	9	eleq1d 2897	. . . . 5 ⊢ (𝑦 ∈ On → ((rec(𝐺, 𝐴)‘suc 𝑦) ∈ 𝑃 ↔ (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃))
11	8, 10	syl5ibr 248	. . . 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 33800	1 ⊢ (𝐶 ∈ ω → (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  ∅c0 4291  Oncon0 6191  suc csuc 6193  ‘cfv 6355  ωcom 7580  reccrdg 8045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046

This theorem is referenced by:  findabrcl  33802