MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eceq1 Structured version   Unicode version

Theorem eceq1 7142
Description: Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995.)
Assertion
Ref Expression
eceq1  |-  ( A  =  B  ->  [ A ] C  =  [ B ] C )

Proof of Theorem eceq1
StepHypRef Expression
1 sneq 3892 . . 3  |-  ( A  =  B  ->  { A }  =  { B } )
21imaeq2d 5174 . 2  |-  ( A  =  B  ->  ( C " { A }
)  =  ( C
" { B }
) )
3 df-ec 7108 . 2  |-  [ A ] C  =  ( C " { A }
)
4 df-ec 7108 . 2  |-  [ B ] C  =  ( C " { B }
)
52, 3, 43eqtr4g 2500 1  |-  ( A  =  B  ->  [ A ] C  =  [ B ] C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   {csn 3882   "cima 4848   [cec 7104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-xp 4851  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ec 7108
This theorem is referenced by:  ecelqsg  7160  snec  7168  qliftfun  7190  qliftfuns  7192  qliftval  7194  ecoptocl  7195  brecop  7198  eroveu  7200  erov  7202  th3qlem1  7211  th3qlem2  7212  th3q  7214  ovec  7215  ecovcom  7216  ecovass  7217  ecovdi  7218  supsrlem  9283  supsr  9284  divsfval  14490  divs0  15744  divsinv  15745  divssub  15746  divsghm  15788  sylow1lem3  16104  sylow2blem2  16125  efgi2  16227  frgpadd  16265  vrgpval  16269  vrgpinv  16271  frgpup3lem  16279  divsabl  16352  divscrng  17327  znzrhval  17984  divstgpopn  19695  divstgplem  19696  elpi1i  20623  pi1addval  20625  pi1xfrf  20630  pi1xfrval  20631  pi1xfrcnvlem  20633  pi1xfrcnv  20634  pi1cof  20636  pi1coval  20637  pi1coghm  20638  vitalilem3  21095  linedegen  28179  fvline  28180  prtlem9  29014  prtlem11  29016
  Copyright terms: Public domain W3C validator