MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imaeq1i Unicode version

Theorem imaeq1i 5025
Description: Equality theorem for image. (Contributed by NM, 21-Dec-2008.)
Hypothesis
Ref Expression
imaeq1i.1  |-  A  =  B
Assertion
Ref Expression
imaeq1i  |-  ( A
" C )  =  ( B " C
)

Proof of Theorem imaeq1i
StepHypRef Expression
1 imaeq1i.1 . 2  |-  A  =  B
2 imaeq1 5023 . 2  |-  ( A  =  B  ->  ( A " C )  =  ( B " C
) )
31, 2ax-mp 8 1  |-  ( A
" C )  =  ( B " C
)
Colors of variables: wff set class
Syntax hints:    = wceq 1632   "cima 4708
This theorem is referenced by:  mptpreima  5182  isarep2  5348  suppfif1  7165  marypha2lem4  7207  dfoi  7242  mapfien  7415  r1limg  7459  isf34lem3  8017  compss  8018  fpwwe2lem13  8280  infmsup  9748  gsumval3  15207  gsumzf1o  15212  gsumzaddlem  15219  dprdfid  15268  evlslem2  16265  ssidcn  17001  cnco  17011  qtopres  17405  idqtop  17413  qtopcn  17421  mbfid  19007  mbfres  19015  cncombf  19029  dvlog  20014  efopnlem2  20020  rinvf1o  23054  disjpreima  23376  mbfmcst  23579  mbfmco  23584  0rrv  23669  areacirclem6  25033  domrancur1c  25305  smbkle  26146  nds  26253  funsnfsup  26865  cytpval  27631
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718
  Copyright terms: Public domain W3C validator