Theorem findabrcl 36667

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 3463	. . . 4 ⊢ (𝐶 ∈ ω → 𝐶 ∈ V)
2		fveq2 6842	. . . . 5 ⊢ (𝑥 = 𝐶 → (rec(𝐺, 𝐴)‘𝑥) = (rec(𝐺, 𝐴)‘𝐶))
3		eqid 2737	. . . . 5 ⊢ (𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥)) = (𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))
4		fvex 6855	. . . . 5 ⊢ (rec(𝐺, 𝐴)‘𝐶) ∈ V
5	2, 3, 4	fvmpt 6949	. . . 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 480	. 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 36666	. . 3 ⊢ (𝐶 ∈ ω → (𝐴 ∈ 𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))
10	9	imp 406	. 2 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ 𝑃) → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃)
11	7, 10	eqeltrd 2837	1 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ 𝑃) → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) ∈ 𝑃)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ↦ cmpt 5181  ‘cfv 6500  ωcom 7818  reccrdg 8350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351

This theorem is referenced by: (None)