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Theorem findabrcl 35643
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 3492 . . . 4 (𝐶 ∈ ω → 𝐶 ∈ V)
2 fveq2 6891 . . . . 5 (𝑥 = 𝐶 → (rec(𝐺, 𝐴)‘𝑥) = (rec(𝐺, 𝐴)‘𝐶))
3 eqid 2731 . . . . 5 (𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥)) = (𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))
4 fvex 6904 . . . . 5 (rec(𝐺, 𝐴)‘𝐶) ∈ V
52, 3, 4fvmpt 6998 . . . 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 480 . 2 ((𝐶 ∈ ω ∧ 𝐴𝑃) → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) = (rec(𝐺, 𝐴)‘𝐶))
8 findabrcl.1 . . . 4 (𝑧𝑃 → (𝐺𝑧) ∈ 𝑃)
98findreccl 35642 . . 3 (𝐶 ∈ ω → (𝐴𝑃 → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃))
109imp 406 . 2 ((𝐶 ∈ ω ∧ 𝐴𝑃) → (rec(𝐺, 𝐴)‘𝐶) ∈ 𝑃)
117, 10eqeltrd 2832 1 ((𝐶 ∈ ω ∧ 𝐴𝑃) → ((𝑥 ∈ V ↦ (rec(𝐺, 𝐴)‘𝑥))‘𝐶) ∈ 𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  Vcvv 3473  cmpt 5231  cfv 6543  ωcom 7859  reccrdg 8413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414
This theorem is referenced by: (None)
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