Theorem suppval 6471

Description: The value of the operation constructing the support of a function. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 6-Apr-2019.)

Assertion

Ref	Expression
suppval	|- ( ( X e. V /\ Z e. W ) -> ( X supp Z ) = { i e. dom X | ( X " { i } ) =/= { Z } } )

Distinct variable groups:   i, X   i, Z

Allowed substitution hints:   V( i)   W( i)

Proof of Theorem suppval

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

Step	Hyp	Ref	Expression
1		df-supp 6470	. . 3 |- supp = ( x e. _V , z e. _V |-> { i e. dom x | ( x " { i } ) =/= { z } } )
2	1	a1i 9	. 2 |- ( ( X e. V /\ Z e. W ) -> supp = ( x e. _V , z e. _V |-> { i e. dom x | ( x " { i } ) =/= { z } } ) )
3		dmeq 4979	. . . . 5 |- ( x = X -> dom x = dom X )
4	3	adantr 276	. . . 4 |- ( ( x = X /\ z = Z ) -> dom x = dom X )
5		imaeq1 5119	. . . . . 6 |- ( x = X -> ( x " { i } ) = ( X " { i } ) )
6	5	adantr 276	. . . . 5 |- ( ( x = X /\ z = Z ) -> ( x " { i } ) = ( X " { i } ) )
7		sneq 3719	. . . . . 6 |- ( z = Z -> { z } = { Z } )
8	7	adantl 277	. . . . 5 |- ( ( x = X /\ z = Z ) -> { z } = { Z } )
9	6, 8	neeq12d 2440	. . . 4 |- ( ( x = X /\ z = Z ) -> ( ( x " { i } ) =/= { z } <-> ( X " { i } ) =/= { Z } ) )
10	4, 9	rabeqbidv 2816	. . 3 |- ( ( x = X /\ z = Z ) -> { i e. dom x | ( x " { i } ) =/= { z } } = { i e. dom X | ( X " { i } ) =/= { Z } } )
11	10	adantl 277	. 2 |- ( ( ( X e. V /\ Z e. W ) /\ ( x = X /\ z = Z ) ) -> { i e. dom x | ( x " { i } ) =/= { z } } = { i e. dom X | ( X " { i } ) =/= { Z } } )
12		elex 2833	. . 3 |- ( X e. V -> X e. _V )
13	12	adantr 276	. 2 |- ( ( X e. V /\ Z e. W ) -> X e. _V )
14		elex 2833	. . 3 |- ( Z e. W -> Z e. _V )
15	14	adantl 277	. 2 |- ( ( X e. V /\ Z e. W ) -> Z e. _V )
16		dmexg 5044	. . . 4 |- ( X e. V -> dom X e. _V )
17	16	adantr 276	. . 3 |- ( ( X e. V /\ Z e. W ) -> dom X e. _V )
18		rabexg 4277	. . 3 |- ( dom X e. _V -> { i e. dom X | ( X " { i } ) =/= { Z } } e. _V )
19	17, 18	syl 14	. 2 |- ( ( X e. V /\ Z e. W ) -> { i e. dom X | ( X " { i } ) =/= { Z } } e. _V )
20	2, 11, 13, 15, 19	ovmpod 6210	1 |- ( ( X e. V /\ Z e. W ) -> ( X supp Z ) = { i e. dom X | ( X " { i } ) =/= { Z } } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   =/= wne 2420   {crab 2532   _Vcvv 2821   {csn 3708   dom cdm 4772   "cima 4775  (class class class)co 6079   e. cmpo 6081   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-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-rab 2537  df-v 2823  df-sbc 3052  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-br 4129  df-opab 4191  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-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-supp 6470

This theorem is referenced by:  supp0  6472  suppval1  6473  suppssdmg  6483  suppsnopdc  6484  ressuppss  6488