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Theorem fvres 5225
Description: The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
Assertion
Ref Expression
fvres (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))

Proof of Theorem fvres
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 2577 . . . . 5 𝑥 ∈ V
21brres 4645 . . . 4 (𝐴(𝐹𝐵)𝑥 ↔ (𝐴𝐹𝑥𝐴𝐵))
32rbaib 841 . . 3 (𝐴𝐵 → (𝐴(𝐹𝐵)𝑥𝐴𝐹𝑥))
43iotabidv 4915 . 2 (𝐴𝐵 → (℩𝑥𝐴(𝐹𝐵)𝑥) = (℩𝑥𝐴𝐹𝑥))
5 df-fv 4937 . 2 ((𝐹𝐵)‘𝐴) = (℩𝑥𝐴(𝐹𝐵)𝑥)
6 df-fv 4937 . 2 (𝐹𝐴) = (℩𝑥𝐴𝐹𝑥)
74, 5, 63eqtr4g 2113 1 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409   class class class wbr 3791  cres 4374  cio 4892  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-xp 4378  df-res 4384  df-iota 4894  df-fv 4937
This theorem is referenced by:  funssfv  5226  feqresmpt  5254  fvreseq  5298  respreima  5322  ffvresb  5355  fnressn  5376  fressnfv  5377  fvresi  5383  fvunsng  5384  fvsnun1  5387  fvsnun2  5388  fsnunfv  5390  funfvima  5417  isoresbr  5476  isores3  5482  isoini2  5485  ovres  5667  ofres  5752  offres  5789  fo1stresm  5815  fo2ndresm  5816  fo2ndf  5875  f1o2ndf1  5876  smores  5937  smores2  5939  tfrlem1  5953  rdgival  5999  rdgon  6003  frec0g  6013  frecsuclem1  6017  frecsuclem2  6019  frecrdg  6022  addpiord  6471  mulpiord  6472  fseq1p1m1  9057  iseqfeq2  9392  shftidt  9661  climres  10054
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