Theorem findreccl 36426

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 8356	. . 3 ⊢ (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘∅) = 𝐴)
2		eleq1a 2823	. . 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 7812	. . . 4 ⊢ (𝑦 ∈ ω → 𝑦 ∈ On)
5		fveq2 6826	. . . . . . 7 ⊢ (𝑧 = (rec(𝐺, 𝐴)‘𝑦) → (𝐺‘𝑧) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
6	5	eleq1d 2813	. . . . . 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 3534	. . . . 5 ⊢ ((rec(𝐺, 𝐴)‘𝑦) ∈ 𝑃 → (𝐺‘(rec(𝐺, 𝐴)‘𝑦)) ∈ 𝑃)
9		rdgsuc 8353	. . . . . 6 ⊢ (𝑦 ∈ On → (rec(𝐺, 𝐴)‘suc 𝑦) = (𝐺‘(rec(𝐺, 𝐴)‘𝑦)))
10	9	eleq1d 2813	. . . . 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 36425	1 ⊢ (𝐶 ∈ ω → (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∅c0 4286  Oncon0 6311  suc csuc 6313  ‘cfv 6486  ωcom 7806  reccrdg 8338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339

This theorem is referenced by:  findabrcl  36427