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Theorem ofmresval 7634
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 7521 . 2 ((𝐹𝐴𝐺𝐵) → (𝐹( ∘f 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹f 𝑅𝐺))
41, 2, 3syl2anc 585 1 (𝜑 → (𝐹( ∘f 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹f 𝑅𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107   × cxp 5632  cres 5636  (class class class)co 7358  f cof 7616
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-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
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-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-res 5646  df-iota 6449  df-fv 6505  df-ov 7361
This theorem is referenced by:  psradd  21353  dchrmul  26599  ldualvadd  37594
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