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Theorem ofmresval 7714
Description: Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
Hypotheses
Ref Expression
ofmresval.f (𝜑𝐹𝐴)
ofmresval.g (𝜑𝐺𝐵)
Assertion
Ref Expression
ofmresval (𝜑 → (𝐹( ∘f 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹f 𝑅𝐺))

Proof of Theorem ofmresval
StepHypRef Expression
1 ofmresval.f . 2 (𝜑𝐹𝐴)
2 ofmresval.g . 2 (𝜑𝐺𝐵)
3 ovres 7600 . 2 ((𝐹𝐴𝐺𝐵) → (𝐹( ∘f 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹f 𝑅𝐺))
41, 2, 3syl2anc 584 1 (𝜑 → (𝐹( ∘f 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹f 𝑅𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107   × cxp 5682  cres 5686  (class class class)co 7432  f cof 7696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-res 5696  df-iota 6513  df-fv 6568  df-ov 7435
This theorem is referenced by:  psradd  21958  dchrmul  27293  ldualvadd  39131
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