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Theorem fvopabf4g 34003
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 8108 . . . 4 ((𝑅𝑌𝐷𝑋) → (𝐴 ∈ (𝑅𝑚 𝐷) ↔ 𝐴:𝐷𝑅))
21ancoms 451 . . 3 ((𝐷𝑋𝑅𝑌) → (𝐴 ∈ (𝑅𝑚 𝐷) ↔ 𝐴:𝐷𝑅))
32biimp3ar 1595 . 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 6507 . 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 198  w3a 1108   = wceq 1653  wcel 2157  Vcvv 3385  cmpt 4922  wf 6097  cfv 6101  (class class class)co 6878  𝑚 cmap 8095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097
This theorem is referenced by: (None)
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