Theorem suppimacnvss 7305

Description: The support of functions "defined" by inverse images is a subset of the support defined by df-supp 7296. (Contributed by AV, 7-Apr-2019.)

Assertion

Ref	Expression
suppimacnvss	|- ( ( R e. V /\ Z e. W ) -> ( `' R " ( _V \ { Z } ) ) C_ ( R supp Z ) )

Proof of Theorem suppimacnvss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		exsimpl 1795	. . . . 5 |- ( E. y ( x R y /\ y =/= Z ) -> E. y x R y )
2		pm5.1 902	. . . . . 6 |- ( ( x R y /\ y =/= Z ) -> ( x R y <-> y =/= Z ) )
3	2	eximi 1762	. . . . 5 |- ( E. y ( x R y /\ y =/= Z ) -> E. y ( x R y <-> y =/= Z ) )
4	1, 3	jca 554	. . . 4 |- ( E. y ( x R y /\ y =/= Z ) -> ( E. y x R y /\ E. y ( x R y <-> y =/= Z ) ) )
5	4	a1i 11	. . 3 |- ( ( R e. V /\ Z e. W ) -> ( E. y ( x R y /\ y =/= Z ) -> ( E. y x R y /\ E. y ( x R y <-> y =/= Z ) ) ) )
6	5	ss2abdv 3675	. 2 |- ( ( R e. V /\ Z e. W ) -> { x | E. y ( x R y /\ y =/= Z ) } C_ { x | ( E. y x R y /\ E. y ( x R y <-> y =/= Z ) ) } )
7		cnvimadfsn 7304	. . 3 |- ( `' R " ( _V \ { Z } ) ) = { x | E. y ( x R y /\ y =/= Z ) }
8	7	a1i 11	. 2 |- ( ( R e. V /\ Z e. W ) -> ( `' R " ( _V \ { Z } ) ) = { x | E. y ( x R y /\ y =/= Z ) } )
9		suppvalbr 7299	. 2 |- ( ( R e. V /\ Z e. W ) -> ( R supp Z ) = { x | ( E. y x R y /\ E. y ( x R y <-> y =/= Z ) ) } )
10	6, 8, 9	3sstr4d 3648	1 |- ( ( R e. V /\ Z e. W ) -> ( `' R " ( _V \ { Z } ) ) C_ ( R supp Z ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   `'ccnv 5113   "cima 5117  (class class class)co 6650   supp csupp 7295

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-8 1992  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  ax-un 6949

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  suppimacnv  7306