Theorem suppval1 6417

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 6415	. . 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 5363	. . . . . . . . 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 5714	. . . . . . 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 2237	. . . . 5 |- ( ( ( Fun X /\ X e. V /\ Z e. W ) /\ i e. dom X ) -> ( X " { i } ) = { ( X ` i ) } )
9	8	neeq1d 2421	. . . 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 2806	. . . . . . . 8 |- i e. _V
12		fvexg 5667	. . . . . . . 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 3851	. . . . . . 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 2444	. . . 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 2791	. 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 2264	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 2202   =/= wne 2403   {crab 2515   _Vcvv 2803   {csn 3673   dom cdm 4731   "cima 4734   Fun wfun 5327   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   supp csupp 6413

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-supp 6414

This theorem is referenced by:  suppvalfng  6418  suppvalfn  6419  suppfnss  6435