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Theorem fvopabf4g 36230
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 8784 . . . 4 ((𝑅𝑌𝐷𝑋) → (𝐴 ∈ (𝑅m 𝐷) ↔ 𝐴:𝐷𝑅))
21ancoms 460 . . 3 ((𝐷𝑋𝑅𝑌) → (𝐴 ∈ (𝑅m 𝐷) ↔ 𝐴:𝐷𝑅))
32biimp3ar 1471 . 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 6952 . 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 205  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3447  cmpt 5192  wf 6496  cfv 6500  (class class class)co 7361  m cmap 8771
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-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-map 8773
This theorem is referenced by: (None)
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