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Theorem ofmresval 7407
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 7299 . 2 ((𝐹𝐴𝐺𝐵) → (𝐹( ∘f 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹f 𝑅𝐺))
41, 2, 3syl2anc 587 1 (𝜑 → (𝐹( ∘f 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹f 𝑅𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114   × cxp 5530  cres 5534  (class class class)co 7140  f cof 7392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-xp 5538  df-res 5544  df-iota 6293  df-fv 6342  df-ov 7143
This theorem is referenced by:  psradd  20618  dchrmul  25830  ldualvadd  36383
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