Theorem suppval1 7301

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 7297	. . 3 |- ( ( X e. V /\ Z e. W ) -> ( X supp Z ) = { i e. dom X | ( X " { i } ) =/= { Z } } )
2	1	3adant1 1079	. 2 |- ( ( Fun X /\ X e. V /\ Z e. W ) -> ( X supp Z ) = { i e. dom X | ( X " { i } ) =/= { Z } } )
3		funfn 5918	. . . . . . . . 9 |- ( Fun X <-> X Fn dom X )
4	3	biimpi 206	. . . . . . . 8 |- ( Fun X -> X Fn dom X )
5	4	3ad2ant1 1082	. . . . . . 7 |- ( ( Fun X /\ X e. V /\ Z e. W ) -> X Fn dom X )
6		fnsnfv 6258	. . . . . . 7 |- ( ( X Fn dom X /\ i e. dom X ) -> { ( X ` i ) } = ( X " { i } ) )
7	5, 6	sylan 488	. . . . . 6 |- ( ( ( Fun X /\ X e. V /\ Z e. W ) /\ i e. dom X ) -> { ( X ` i ) } = ( X " { i } ) )
8	7	eqcomd 2628	. . . . 5 |- ( ( ( Fun X /\ X e. V /\ Z e. W ) /\ i e. dom X ) -> ( X " { i } ) = { ( X ` i ) } )
9	8	neeq1d 2853	. . . 4 |- ( ( ( Fun X /\ X e. V /\ Z e. W ) /\ i e. dom X ) -> ( ( X " { i } ) =/= { Z } <-> { ( X ` i ) } =/= { Z } ) )
10		fvex 6201	. . . . . 6 |- ( X ` i ) e. _V
11		sneqbg 4374	. . . . . 6 |- ( ( X ` i ) e. _V -> ( { ( X ` i ) } = { Z } <-> ( X ` i ) = Z ) )
12	10, 11	mp1i 13	. . . . 5 |- ( ( ( Fun X /\ X e. V /\ Z e. W ) /\ i e. dom X ) -> ( { ( X ` i ) } = { Z } <-> ( X ` i ) = Z ) )
13	12	necon3bid 2838	. . . 4 |- ( ( ( Fun X /\ X e. V /\ Z e. W ) /\ i e. dom X ) -> ( { ( X ` i ) } =/= { Z } <-> ( X ` i ) =/= Z ) )
14	9, 13	bitrd 268	. . 3 |- ( ( ( Fun X /\ X e. V /\ Z e. W ) /\ i e. dom X ) -> ( ( X " { i } ) =/= { Z } <-> ( X ` i ) =/= Z ) )
15	14	rabbidva 3188	. 2 |- ( ( Fun X /\ X e. V /\ Z e. W ) -> { i e. dom X | ( X " { i } ) =/= { Z } } = { i e. dom X | ( X ` i ) =/= Z } )
16	2, 15	eqtrd 2656	1 |- ( ( Fun X /\ X e. V /\ Z e. W ) -> ( X supp Z ) = { i e. dom X | ( X ` i ) =/= Z } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   {csn 4177   dom cdm 5114   "cima 5117   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   supp csupp 7295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  suppvalfn  7302  suppfnss  7320  fnsuppres  7322  domnmsuppn0  42150  rmsuppss  42151  mndpsuppss  42152  scmsuppss  42153  suppdm  42300