Theorem compss 9198

Description: Express image under of the complementation isomorphism. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)

Hypothesis

Ref	Expression
compss.a	|- F = ( x e. ~P A |-> ( A \ x ) )

Assertion

Ref	Expression
compss	|- ( F " G ) = { y e. ~P A | ( A \ y ) e. G }

Distinct variable groups:   x, y, A   y, F   y, G

Allowed substitution hints:   F( x)   G( x)

Proof of Theorem compss

Step	Hyp	Ref	Expression
1		compss.a	. . . 4 |- F = ( x e. ~P A |-> ( A \ x ) )
2	1	compsscnv 9193	. . 3 |- `' F = F
3	2	imaeq1i 5463	. 2 |- ( `' F " G ) = ( F " G )
4		difeq2 3722	. . . . 5 |- ( x = y -> ( A \ x ) = ( A \ y ) )
5	4	cbvmptv 4750	. . . 4 |- ( x e. ~P A |-> ( A \ x ) ) = ( y e. ~P A |-> ( A \ y ) )
6	1, 5	eqtri 2644	. . 3 |- F = ( y e. ~P A |-> ( A \ y ) )
7	6	mptpreima 5628	. 2 |- ( `' F " G ) = { y e. ~P A | ( A \ y ) e. G }
8	3, 7	eqtr3i 2646	1 |- ( F " G ) = { y e. ~P A | ( A \ y ) e. G }

Syntax hints:   = wceq 1483   e. wcel 1990   {crab 2916   \ cdif 3571   ~Pcpw 4158   |-> cmpt 4729   `'ccnv 5113   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  isf34lem4  9199