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Theorem difpreima 5294
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 4958 . 2  |-  ( Fun 
F  ->  Fun  `' `' F )
2 imadif 4979 . 2  |-  ( Fun  `' `' F  ->  ( `' F " ( A 
\  B ) )  =  ( ( `' F " A ) 
\  ( `' F " B ) ) )
31, 2syl 14 1  |-  ( Fun 
F  ->  ( `' F " ( A  \  B ) )  =  ( ( `' F " A )  \  ( `' F " B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    \ cdif 2914   `'ccnv 4344   "cima 4348   Fun wfun 4896
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-in1 544  ax-in2 545  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-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fun 4904
This theorem is referenced by: (None)
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