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Theorem fvopabf4g 35107
 Description: Function value of an operator abstraction whose domain is a set of functions with given domain and range. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
Hypotheses
Ref Expression
fvopabf4g.1 𝐶 ∈ V
fvopabf4g.2 (𝑥 = 𝐴𝐵 = 𝐶)
fvopabf4g.3 𝐹 = (𝑥 ∈ (𝑅m 𝐷) ↦ 𝐵)
Assertion
Ref Expression
fvopabf4g ((𝐷𝑋𝑅𝑌𝐴:𝐷𝑅) → (𝐹𝐴) = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷   𝑥,𝑅
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)   𝑋(𝑥)   𝑌(𝑥)

Proof of Theorem fvopabf4g
StepHypRef Expression
1 elmapg 8415 . . . 4 ((𝑅𝑌𝐷𝑋) → (𝐴 ∈ (𝑅m 𝐷) ↔ 𝐴:𝐷𝑅))
21ancoms 462 . . 3 ((𝐷𝑋𝑅𝑌) → (𝐴 ∈ (𝑅m 𝐷) ↔ 𝐴:𝐷𝑅))
32biimp3ar 1467 . 2 ((𝐷𝑋𝑅𝑌𝐴:𝐷𝑅) → 𝐴 ∈ (𝑅m 𝐷))
4 fvopabf4g.2 . . 3 (𝑥 = 𝐴𝐵 = 𝐶)
5 fvopabf4g.3 . . 3 𝐹 = (𝑥 ∈ (𝑅m 𝐷) ↦ 𝐵)
6 fvopabf4g.1 . . 3 𝐶 ∈ V
74, 5, 6fvmpt 6759 . 2 (𝐴 ∈ (𝑅m 𝐷) → (𝐹𝐴) = 𝐶)
83, 7syl 17 1 ((𝐷𝑋𝑅𝑌𝐴:𝐷𝑅) → (𝐹𝐴) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  Vcvv 3480   ↦ cmpt 5132  ⟶wf 6339  ‘cfv 6343  (class class class)co 7149   ↑m cmap 8402 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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404 This theorem is referenced by: (None)
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