Theorem findreccl 32577

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 7568	. . 3 ⊢ (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘∅) = 𝐴)
2		eleq1a 2725	. . 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 7113	. . . 4 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
5		fveq2 6229	. . . . . . 7 ⊢ (𝑧 = (rec(𝐺, 𝐴)‘𝑦) → (𝐺‘𝑧) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
6	5	eleq1d 2715	. . . . . 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 3303	. . . . 5 ⊢ ((rec(𝐺, 𝐴)‘𝑦) ∈ 𝑃 → (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃)
9		rdgsuc 7565	. . . . . 6 ⊢ (𝑦 ∈ On → (rec(𝐺, 𝐴)‘suc 𝑦) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
10	9	eleq1d 2715	. . . . 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 32576	1 ⊢ (𝐶 ∈ ω → (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))

Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  ∅c0 3948  Oncon0 5761  suc csuc 5763  ‘cfv 5926  ωcom 7107  reccrdg 7550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551

This theorem is referenced by:  findabrcl  32578