Theorem suppval 6450

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 6449	. . 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 4961	. . . . 5 |- ( x = X -> dom x = dom X )
4	3	adantr 276	. . . 4 |- ( ( x = X /\ z = Z ) -> dom x = dom X )
5		imaeq1 5101	. . . . . 6 |- ( x = X -> ( x " { i } ) = ( X " { i } ) )
6	5	adantr 276	. . . . 5 |- ( ( x = X /\ z = Z ) -> ( x " { i } ) = ( X " { i } ) )
7		sneq 3705	. . . . . 6 |- ( z = Z -> { z } = { Z } )
8	7	adantl 277	. . . . 5 |- ( ( x = X /\ z = Z ) -> { z } = { Z } )
9	6, 8	neeq12d 2434	. . . 4 |- ( ( x = X /\ z = Z ) -> ( ( x " { i } ) =/= { z } <-> ( X " { i } ) =/= { Z } ) )
10	4, 9	rabeqbidv 2810	. . 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 2827	. . 3 |- ( X e. V -> X e. _V )
13	12	adantr 276	. 2 |- ( ( X e. V /\ Z e. W ) -> X e. _V )
14		elex 2827	. . 3 |- ( Z e. W -> Z e. _V )
15	14	adantl 277	. 2 |- ( ( X e. V /\ Z e. W ) -> Z e. _V )
16		dmexg 5026	. . . 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 4260	. . 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 6189	1 |- ( ( X e. V /\ Z e. W ) -> ( X supp Z ) = { i e. dom X | ( X " { i } ) =/= { Z } } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   =/= wne 2414   {crab 2526   _Vcvv 2815   {csn 3694   dom cdm 4754   "cima 4757  (class class class)co 6058   e. cmpo 6060   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-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-rab 2531  df-v 2817  df-sbc 3046  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-br 4115  df-opab 4177  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-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-supp 6449

This theorem is referenced by:  supp0  6451  suppval1  6452  suppssdmg  6462  suppsnopdc  6463  ressuppss  6467