Theorem difpreima 5761

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

Step	Hyp	Ref	Expression
1		funcnvcnv 5379	. 2 |- ( Fun F -> Fun `' `' F )
2		imadif 5400	. 2 |- ( Fun `' `' F -> ( `' F " ( A \ B ) ) = ( ( `' F " A ) \ ( `' F " B ) ) )
3	1, 2	syl 14	1 |- ( Fun F -> ( `' F " ( A \ B ) ) = ( ( `' F " A ) \ ( `' F " B ) ) )

Syntax hints:   -> wi 4   = wceq 1395   \ cdif 3194   `'ccnv 4717   "cima 4721   Fun wfun 5311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-fun 5319

This theorem is referenced by: (None)