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Theorem findabrcl 35277
Description: Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
findabrcl.1 (𝑧𝑃 → (𝐺𝑧) ∈ 𝑃)
Assertion
Ref Expression
findabrcl ((𝐶 ∈ ω ∧ 𝐴𝑃) → ((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
StepHypRef Expression
1 elex 3493 . . . 4 (𝐶 ∈ ω → 𝐶 ∈ V)
2 fveq2 6888 . . . . 5 (𝑥 = 𝐶 → (rec(𝐺, 𝐴)‘𝑥) = (rec(𝐺, 𝐴)‘𝐶))
3 eqid 2733 . . . . 5 (𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥)) = (𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))
4 fvex 6901 . . . . 5 (rec(𝐺, 𝐴)‘𝐶) ∈ V
52, 3, 4fvmpt 6994 . . . 4 (𝐶 ∈ V → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) = (rec(𝐺, 𝐴)‘𝐶))
61, 5syl 17 . . 3 (𝐶 ∈ ω → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) = (rec(𝐺, 𝐴)‘𝐶))
76adantr 482 . 2 ((𝐶 ∈ ω ∧ 𝐴𝑃) → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) = (rec(𝐺, 𝐴)‘𝐶))
8 findabrcl.1 . . . 4 (𝑧𝑃 → (𝐺𝑧) ∈ 𝑃)
98findreccl 35276 . . 3 (𝐶 ∈ ω → (𝐴𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))
109imp 408 . 2 ((𝐶 ∈ ω ∧ 𝐴𝑃) → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃)
117, 10eqeltrd 2834 1 ((𝐶 ∈ ω ∧ 𝐴𝑃) → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) ∈ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  cmpt 5230  cfv 6540  ωcom 7850  reccrdg 8404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405
This theorem is referenced by: (None)
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