Theorem mptsuppd 6455

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 6454	. 2 |- ( ph -> ( F supp Z ) = { x e. A | B e. ( _V \ { Z } ) } )
5		eldifsn 3819	. . . 4 |- ( B e. ( _V \ { Z } ) <-> ( B e. _V /\ B =/= Z ) )
6		mptsuppd.b	. . . . . 6 |- ( ( ph /\ x e. A ) -> B e. U )
7	6	elexd 2826	. . . . 5 |- ( ( ph /\ x e. A ) -> B e. _V )
8	7	biantrurd 305	. . . 4 |- ( ( ph /\ x e. A ) -> ( B =/= Z <-> ( B e. _V /\ B =/= Z ) ) )
9	5, 8	bitr4id 199	. . 3 |- ( ( ph /\ x e. A ) -> ( B e. ( _V \ { Z } ) <-> B =/= Z ) )
10	9	rabbidva 2800	. 2 |- ( ph -> { x e. A | B e. ( _V \ { Z } ) } = { x e. A | B =/= Z } )
11	4, 10	eqtrd 2265	1 |- ( ph -> ( F supp Z ) = { x e. A | B =/= Z } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   =/= wne 2412   {crab 2524   _Vcvv 2812   \ cdif 3207   {csn 3688   |-> cmpt 4170  (class class class)co 6049   supp csupp 6434

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-supp 6435

This theorem is referenced by: (None)