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Theorem fvopabf4g 33847
 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 8022 . . . 4 ((𝑅𝑌𝐷𝑋) → (𝐴 ∈ (𝑅𝑚 𝐷) ↔ 𝐴:𝐷𝑅))
21ancoms 455 . . 3 ((𝐷𝑋𝑅𝑌) → (𝐴 ∈ (𝑅𝑚 𝐷) ↔ 𝐴:𝐷𝑅))
32biimp3ar 1581 . 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 6424 . 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 196   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  Vcvv 3351   ↦ cmpt 4863  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793   ↑𝑚 cmap 8009 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011 This theorem is referenced by: (None)
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