MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reseq12i Structured version   Unicode version

Theorem reseq12i 5123
Description: Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
Hypotheses
Ref Expression
reseqi.1  |-  A  =  B
reseqi.2  |-  C  =  D
Assertion
Ref Expression
reseq12i  |-  ( A  |`  C )  =  ( B  |`  D )

Proof of Theorem reseq12i
StepHypRef Expression
1 reseqi.1 . . 3  |-  A  =  B
21reseq1i 5121 . 2  |-  ( A  |`  C )  =  ( B  |`  C )
3 reseqi.2 . . 3  |-  C  =  D
43reseq2i 5122 . 2  |-  ( B  |`  C )  =  ( B  |`  D )
52, 4eqtri 2458 1  |-  ( A  |`  C )  =  ( B  |`  D )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    |` cres 4856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089  df-in 3449  df-opab 4485  df-xp 4860  df-res 4866
This theorem is referenced by:  cnvresid  5671  wfrlem5  7048  dfoi  8026  lubfval  16175  glbfval  16188  oduglb  16336  odulub  16338  dvlog  23461  dvlog2  23463  sitgclg  29003  frrlem5  30305  fourierdlem57  37595  fourierdlem74  37612  fourierdlem75  37613
  Copyright terms: Public domain W3C validator