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

Theorem difpreima 5947
Description: Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
Assertion
Ref Expression
difpreima  |-  ( Fun 
F  ->  ( `' F " ( A  \  B ) )  =  ( ( `' F " A )  \  ( `' F " B ) ) )

Proof of Theorem difpreima
StepHypRef Expression
1 funcnvcnv 5581 . 2  |-  ( Fun 
F  ->  Fun  `' `' F )
2 imadif 5598 . 2  |-  ( Fun  `' `' F  ->  ( `' F " ( A 
\  B ) )  =  ( ( `' F " A ) 
\  ( `' F " B ) ) )
31, 2syl 17 1  |-  ( Fun 
F  ->  ( `' F " ( A  \  B ) )  =  ( ( `' F " A )  \  ( `' F " B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1403    \ cdif 3408   `'ccnv 4939   "cima 4943   Fun wfun 5517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-br 4393  df-opab 4451  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-fun 5525
This theorem is referenced by:  gsumpropd2lem  16114  fsumcvg4  28266  zrhunitpreima  28292  imambfm  28591  carsggect  28647  sibfof  28669  eulerpartlemmf  28701  dvtanlemOLD  31401  itg2addnclem  31403  itg2addnclem2  31404
  Copyright terms: Public domain W3C validator