ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  imaeq12d Unicode version

Theorem imaeq12d 4669
Description: Equality theorem for image. (Contributed by Mario Carneiro, 4-Dec-2016.)
Hypotheses
Ref Expression
imaeq1d.1  |-  ( ph  ->  A  =  B )
imaeq12d.2  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
imaeq12d  |-  ( ph  ->  ( A " C
)  =  ( B
" D ) )

Proof of Theorem imaeq12d
StepHypRef Expression
1 imaeq1d.1 . . 3  |-  ( ph  ->  A  =  B )
21imaeq1d 4667 . 2  |-  ( ph  ->  ( A " C
)  =  ( B
" C ) )
3 imaeq12d.2 . . 3  |-  ( ph  ->  C  =  D )
43imaeq2d 4668 . 2  |-  ( ph  ->  ( B " C
)  =  ( B
" D ) )
52, 4eqtrd 2072 1  |-  ( ph  ->  ( A " C
)  =  ( B
" D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243   "cima 4348
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358
This theorem is referenced by:  csbima12g  4686
  Copyright terms: Public domain W3C validator