Theorem findabrcl 34329

Description: Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)

Hypothesis

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

Assertion

Ref	Expression
findabrcl	⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ 𝑃) → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) ∈ 𝑃)

Distinct variable groups:   𝑥,𝐺   𝑥,𝐴   𝑥,𝐶   𝑧,𝐺   𝑧,𝐴   𝑧,𝑃

Allowed substitution hints:   𝐶(𝑧)   𝑃(𝑥)

Proof of Theorem findabrcl

Step	Hyp	Ref	Expression
1		elex 3416	. . . 4 ⊢ (𝐶 ∈ ω → 𝐶 ∈ V)
2		fveq2 6695	. . . . 5 ⊢ (𝑥 = 𝐶 → (rec(𝐺, 𝐴)‘𝑥) = (rec(𝐺, 𝐴)‘𝐶))
3		eqid 2736	. . . . 5 ⊢ (𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥)) = (𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))
4		fvex 6708	. . . . 5 ⊢ (rec(𝐺, 𝐴)‘𝐶) ∈ V
5	2, 3, 4	fvmpt 6796	. . . 4 ⊢ (𝐶 ∈ V → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) = (rec(𝐺, 𝐴)‘𝐶))
6	1, 5	syl 17	. . 3 ⊢ (𝐶 ∈ ω → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) = (rec(𝐺, 𝐴)‘𝐶))
7	6	adantr 484	. 2 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ 𝑃) → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) = (rec(𝐺, 𝐴)‘𝐶))
8		findabrcl.1	. . . 4 ⊢ (𝑧 ∈ 𝑃 → (𝐺‘𝑧) ∈ 𝑃)
9	8	findreccl 34328	. . 3 ⊢ (𝐶 ∈ ω → (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))
10	9	imp 410	. 2 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ 𝑃) → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃)
11	7, 10	eqeltrd 2831	1 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ 𝑃) → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) ∈ 𝑃)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2112  Vcvv 3398   ↦ cmpt 5120  ‘cfv 6358  ωcom 7622  reccrdg 8123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-om 7623  df-wrecs 8025  df-recs 8086  df-rdg 8124

This theorem is referenced by: (None)