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Theorem reseq12d 5150
Description: Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.)
Hypotheses
Ref Expression
reseqd.1  |-  ( ph  ->  A  =  B )
reseqd.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
reseq12d  |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  D ) )

Proof of Theorem reseq12d
StepHypRef Expression
1 reseqd.1 . . 3  |-  ( ph  ->  A  =  B )
21reseq1d 5148 . 2  |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  C ) )
3 reseqd.2 . . 3  |-  ( ph  ->  C  =  D )
43reseq2d 5149 . 2  |-  ( ph  ->  ( B  |`  C )  =  ( B  |`  D ) )
52, 4eqtrd 2470 1  |-  ( ph  ->  ( A  |`  C )  =  ( B  |`  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    |` cres 4883
This theorem is referenced by:  oieq1  7484  oieq2  7485  ackbij2lem3  8126  setsvalg  13497  resfval  14094  resfval2  14095  resf2nd  14097  dpjfval  15618  psrval  16434  znval  16821  prdsdsf  18402  prdsxmet  18404  imasdsf1olem  18408  xpsxmetlem  18414  xpsmet  18417  isxms  18482  isms  18484  setsxms  18514  setsms  18515  ressxms  18560  ressms  18561  prdsxmslem2  18564  iscms  19303  cmsss  19308  minveclem3a  19333  dvcmulf  19836  efcvx  20370  ispth  21573  constr3pthlem1  21647  sitgclcn  24663  sitgclre  24664  prdsbnd2  26518  cnpwstotbnd  26520  dfateq12d  27983  ldualset  29997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329  df-opab 4270  df-xp 4887  df-res 4893
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