Theorem difpreima 5770

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 5386	. 2 |- ( Fun F -> Fun `' `' F )
2		imadif 5407	. 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 3195   `'ccnv 4722   "cima 4726   Fun wfun 5318

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 4205  ax-pow 4262  ax-pr 4297

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 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-fun 5326

This theorem is referenced by: (None)