Theorem mptsuppd 7318

Description: The support of a function in maps-to notation. (Contributed by AV, 10-Apr-2019.) (Revised by AV, 28-May-2019.)

Hypotheses

Ref	Expression
mptsuppdifd.f	|- F = ( x e. A |-> B )
mptsuppdifd.a	|- ( ph -> A e. V )
mptsuppdifd.z	|- ( ph -> Z e. W )
mptsuppd.b	|- ( ( ph /\ x e. A ) -> B e. U )

Assertion

Ref	Expression
mptsuppd	|- ( ph -> ( F supp Z ) = { x e. A | B =/= Z } )

Distinct variable groups:   x, A   x, Z   ph, x

Allowed substitution hints:   B( x)   U( x)   F( x)   V( x)   W( x)

Proof of Theorem mptsuppd

Step	Hyp	Ref	Expression
1		mptsuppdifd.f	. . 3 |- F = ( x e. A |-> B )
2		mptsuppdifd.a	. . 3 |- ( ph -> A e. V )
3		mptsuppdifd.z	. . 3 |- ( ph -> Z e. W )
4	1, 2, 3	mptsuppdifd 7317	. 2 |- ( ph -> ( F supp Z ) = { x e. A | B e. ( _V \ { Z } ) } )
5		mptsuppd.b	. . . . . 6 |- ( ( ph /\ x e. A ) -> B e. U )
6		elex 3212	. . . . . 6 |- ( B e. U -> B e. _V )
7	5, 6	syl 17	. . . . 5 |- ( ( ph /\ x e. A ) -> B e. _V )
8	7	biantrurd 529	. . . 4 |- ( ( ph /\ x e. A ) -> ( B =/= Z <-> ( B e. _V /\ B =/= Z ) ) )
9		eldifsn 4317	. . . 4 |- ( B e. ( _V \ { Z } ) <-> ( B e. _V /\ B =/= Z ) )
10	8, 9	syl6rbbr 279	. . 3 |- ( ( ph /\ x e. A ) -> ( B e. ( _V \ { Z } ) <-> B =/= Z ) )
11	10	rabbidva 3188	. 2 |- ( ph -> { x e. A | B e. ( _V \ { Z } ) } = { x e. A | B =/= Z } )
12	4, 11	eqtrd 2656	1 |- ( ph -> ( F supp Z ) = { x e. A | B =/= Z } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   \ cdif 3571   {csn 4177   |-> cmpt 4729  (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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  rmsupp0  42149  domnmsuppn0  42150  rmsuppss  42151  suppmptcfin  42160  lcoc0  42211  linc1  42214