Theorem findreccl 36455

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 8468	. . 3 ⊢ (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘∅) = 𝐴)
2		eleq1a 2835	. . 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 7894	. . . 4 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
5		fveq2 6905	. . . . . . 7 ⊢ (𝑧 = (rec(𝐺, 𝐴)‘𝑦) → (𝐺‘𝑧) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
6	5	eleq1d 2825	. . . . . 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 3576	. . . . 5 ⊢ ((rec(𝐺, 𝐴)‘𝑦) ∈ 𝑃 → (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃)
9		rdgsuc 8465	. . . . . 6 ⊢ (𝑦 ∈ On → (rec(𝐺, 𝐴)‘suc 𝑦) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
10	9	eleq1d 2825	. . . . 5 ⊢ (𝑦 ∈ On → ((rec(𝐺, 𝐴)‘suc 𝑦) ∈ 𝑃 ↔ (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃))
11	8, 10	imbitrrid 246	. . . 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 36454	1 ⊢ (𝐶 ∈ ω → (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ∅c0 4332  Oncon0 6383  suc csuc 6385  ‘cfv 6560  ωcom 7888  reccrdg 8450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451

This theorem is referenced by:  findabrcl  36456