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

Theorem fvresd 6246
Description: The value of a restricted function, deduction version of fvres 6245. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypothesis
Ref Expression
fvresd.1 (𝜑𝐴𝐵)
Assertion
Ref Expression
fvresd (𝜑 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))

Proof of Theorem fvresd
StepHypRef Expression
1 fvresd.1 . 2 (𝜑𝐴𝐵)
2 fvres 6245 . 2 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
31, 2syl 17 1 (𝜑 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  cres 5145  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-res 5155  df-iota 5889  df-fv 5934
This theorem is referenced by:  ackbij2lem2  9100  cfsmolem  9130  txkgen  21503  loglesqrt  24544  uhgrspansubgrlem  26227  wlkres  26623  ftc2re  30804  reprsuc  30821  frrlem4  31908  nolesgn2o  31949  nolesgn2ores  31950  noresle  31971  noprefixmo  31973  nosupres  31978  nosupbnd2lem1  31986  noetalem3  31990  limsupresxr  40316  liminfresxr  40317  sssmf  41268
  Copyright terms: Public domain W3C validator