MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ofmresval Structured version   Visualization version   GIF version

Theorem ofmresval 6875
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 6765 . 2 ((𝐹𝐴𝐺𝐵) → (𝐹( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹𝑓 𝑅𝐺))
41, 2, 3syl2anc 692 1 (𝜑 → (𝐹( ∘𝑓 𝑅 ↾ (𝐴 × 𝐵))𝐺) = (𝐹𝑓 𝑅𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987   × cxp 5082  cres 5086  (class class class)co 6615  𝑓 cof 6860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-res 5096  df-iota 5820  df-fv 5865  df-ov 6618
This theorem is referenced by:  psradd  19322  dchrmul  24907  ldualvadd  33935
  Copyright terms: Public domain W3C validator