Theorem findabrcl 36437

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 3471	. . . 4 ⊢ (𝐶 ∈ ω → 𝐶 ∈ V)
2		fveq2 6860	. . . . 5 ⊢ (𝑥 = 𝐶 → (rec(𝐺, 𝐴)‘𝑥) = (rec(𝐺, 𝐴)‘𝐶))
3		eqid 2730	. . . . 5 ⊢ (𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥)) = (𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))
4		fvex 6873	. . . . 5 ⊢ (rec(𝐺, 𝐴)‘𝐶) ∈ V
5	2, 3, 4	fvmpt 6970	. . . 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 36436	. . 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 2829	1 ⊢ ((𝐶 ∈ ω ∧ 𝐴 ∈ 𝑃) → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) ∈ 𝑃)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ↦ cmpt 5190  ‘cfv 6513  ωcom 7844  reccrdg 8379

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 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pr 5389  ax-un 7713

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-ov 7392  df-om 7845  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380

This theorem is referenced by: (None)