Theorem suppval1 6469

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 6467	. . 3 |- ( ( X e. V /\ Z e. W ) -> ( X supp Z ) = { i e. dom X | ( X " { i } ) =/= { Z } } )
2	1	3adant1 1046	. 2 |- ( ( Fun X /\ X e. V /\ Z e. W ) -> ( X supp Z ) = { i e. dom X | ( X " { i } ) =/= { Z } } )
3		funfn 5402	. . . . . . . . 9 |- ( Fun X <-> X Fn dom X )
4	3	biimpi 120	. . . . . . . 8 |- ( Fun X -> X Fn dom X )
5	4	3ad2ant1 1049	. . . . . . 7 |- ( ( Fun X /\ X e. V /\ Z e. W ) -> X Fn dom X )
6		fnsnfv 5756	. . . . . . 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 2244	. . . . 5 |- ( ( ( Fun X /\ X e. V /\ Z e. W ) /\ i e. dom X ) -> ( X " { i } ) = { ( X ` i ) } )
9	8	neeq1d 2438	. . . 4 |- ( ( ( Fun X /\ X e. V /\ Z e. W ) /\ i e. dom X ) -> ( ( X " { i } ) =/= { Z } <-> { ( X ` i ) } =/= { Z } ) )
10		simp2 1029	. . . . . . . 8 |- ( ( Fun X /\ X e. V /\ Z e. W ) -> X e. V )
11		vex 2824	. . . . . . . 8 |- i e. _V
12		fvexg 5709	. . . . . . . 8 |- ( ( X e. V /\ i e. _V ) -> ( X ` i ) e. _V )
13	10, 11, 12	sylancl 417	. . . . . . 7 |- ( ( Fun X /\ X e. V /\ Z e. W ) -> ( X ` i ) e. _V )
14		sneqbg 3883	. . . . . . 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 2461	. . . 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 2809	. 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 2271	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 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   {crab 2532   _Vcvv 2821   {csn 3705   dom cdm 4769   "cima 4772   Fun wfun 5366   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   supp csupp 6465

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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-supp 6466

This theorem is referenced by:  suppvalfng  6470  suppvalfn  6471  suppfnss  6487