Theorem mptsuppd 6469

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 6468	. 2 |- ( ph -> ( F supp Z ) = { x e. A | B e. ( _V \ { Z } ) } )
5		eldifsn 3825	. . . 4 |- ( B e. ( _V \ { Z } ) <-> ( B e. _V /\ B =/= Z ) )
6		mptsuppd.b	. . . . . 6 |- ( ( ph /\ x e. A ) -> B e. U )
7	6	elexd 2829	. . . . 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 2803	. 2 |- ( ph -> { x e. A | B e. ( _V \ { Z } ) } = { x e. A | B =/= Z } )
11	4, 10	eqtrd 2267	1 |- ( ph -> ( F supp Z ) = { x e. A | B =/= Z } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   =/= wne 2414   {crab 2526   _Vcvv 2815   \ cdif 3211   {csn 3694   |-> cmpt 4176  (class class class)co 6058   supp csupp 6448

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-supp 6449

This theorem is referenced by: (None)