Theorem mptsuppd 6490

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 6489	. 2 |- ( ph -> ( F supp Z ) = { x e. A | B e. ( _V \ { Z } ) } )
5		eldifsn 3839	. . . 4 |- ( B e. ( _V \ { Z } ) <-> ( B e. _V /\ B =/= Z ) )
6		mptsuppd.b	. . . . . 6 |- ( ( ph /\ x e. A ) -> B e. U )
7	6	elexd 2835	. . . . 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 2809	. 2 |- ( ph -> { x e. A | B e. ( _V \ { Z } ) } = { x e. A | B =/= Z } )
11	4, 10	eqtrd 2271	1 |- ( ph -> ( F supp Z ) = { x e. A | B =/= Z } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   =/= wne 2420   {crab 2532   _Vcvv 2821   \ cdif 3217   {csn 3708   |-> cmpt 4190  (class class class)co 6079   supp csupp 6469

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-supp 6470

This theorem is referenced by: (None)