Theorem suppval1 6452

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

Assertion

Ref	Expression
suppval1	|- ( ( Fun X /\ X e. V /\ Z e. W ) -> ( X supp Z ) = { i e. dom X | ( X ` i ) =/= Z } )

Distinct variable groups:   i, V   i, W   i, X   i, Z

Proof of Theorem suppval1

Step	Hyp	Ref	Expression
1		suppval 6450	. . 3 |- ( ( X e. V /\ Z e. W ) -> ( X supp Z ) = { i e. dom X | ( X " { i } ) =/= { Z } } )
2	1	3adant1 1042	. 2 |- ( ( Fun X /\ X e. V /\ Z e. W ) -> ( X supp Z ) = { i e. dom X | ( X " { i } ) =/= { Z } } )
3		funfn 5387	. . . . . . . . 9 |- ( Fun X <-> X Fn dom X )
4	3	biimpi 120	. . . . . . . 8 |- ( Fun X -> X Fn dom X )
5	4	3ad2ant1 1045	. . . . . . 7 |- ( ( Fun X /\ X e. V /\ Z e. W ) -> X Fn dom X )
6		fnsnfv 5741	. . . . . . 7 |- ( ( X Fn dom X /\ i e. dom X ) -> { ( X ` i ) } = ( X " { i } ) )
7	5, 6	sylan 283	. . . . . 6 |- ( ( ( Fun X /\ X e. V /\ Z e. W ) /\ i e. dom X ) -> { ( X ` i ) } = ( X " { i } ) )
8	7	eqcomd 2240	. . . . 5 |- ( ( ( Fun X /\ X e. V /\ Z e. W ) /\ i e. dom X ) -> ( X " { i } ) = { ( X ` i ) } )
9	8	neeq1d 2432	. . . 4 |- ( ( ( Fun X /\ X e. V /\ Z e. W ) /\ i e. dom X ) -> ( ( X " { i } ) =/= { Z } <-> { ( X ` i ) } =/= { Z } ) )
10		simp2 1025	. . . . . . . 8 |- ( ( Fun X /\ X e. V /\ Z e. W ) -> X e. V )
11		vex 2818	. . . . . . . 8 |- i e. _V
12		fvexg 5694	. . . . . . . 8 |- ( ( X e. V /\ i e. _V ) -> ( X ` i ) e. _V )
13	10, 11, 12	sylancl 413	. . . . . . 7 |- ( ( Fun X /\ X e. V /\ Z e. W ) -> ( X ` i ) e. _V )
14		sneqbg 3872	. . . . . . 7 |- ( ( X ` i ) e. _V -> ( { ( X ` i ) } = { Z } <-> ( X ` i ) = Z ) )
15	13, 14	syl 14	. . . . . 6 |- ( ( Fun X /\ X e. V /\ Z e. W ) -> ( { ( X ` i ) } = { Z } <-> ( X ` i ) = Z ) )
16	15	adantr 276	. . . . 5 |- ( ( ( Fun X /\ X e. V /\ Z e. W ) /\ i e. dom X ) -> ( { ( X ` i ) } = { Z } <-> ( X ` i ) = Z ) )
17	16	necon3bid 2455	. . . 4 |- ( ( ( Fun X /\ X e. V /\ Z e. W ) /\ i e. dom X ) -> ( { ( X ` i ) } =/= { Z } <-> ( X ` i ) =/= Z ) )
18	9, 17	bitrd 188	. . 3 |- ( ( ( Fun X /\ X e. V /\ Z e. W ) /\ i e. dom X ) -> ( ( X " { i } ) =/= { Z } <-> ( X ` i ) =/= Z ) )
19	18	rabbidva 2803	. 2 |- ( ( Fun X /\ X e. V /\ Z e. W ) -> { i e. dom X | ( X " { i } ) =/= { Z } } = { i e. dom X | ( X ` i ) =/= Z } )
20	2, 19	eqtrd 2267	1 |- ( ( Fun X /\ X e. V /\ Z e. W ) -> ( X supp Z ) = { i e. dom X | ( X ` i ) =/= Z } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   {crab 2526   _Vcvv 2815   {csn 3694   dom cdm 4754   "cima 4757   Fun wfun 5351   Fn wfn 5352   ` cfv 5357  (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-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-fn 5360  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-supp 6449

This theorem is referenced by:  suppvalfng  6453  suppvalfn  6454  suppfnss  6470