Theorem fczsupp0 6493

Description: The support of a constant function with value zero is empty. (Contributed by AV, 30-Jun-2019.)

Assertion

Ref	Expression
fczsupp0	|- ( ( B X. { Z } ) supp Z ) = (/)

Proof of Theorem fczsupp0

Dummy variables x f q z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-supp 6470	. . . . . 6 |- supp = ( f e. _V , z e. _V |-> { q e. dom f | ( f " { q } ) =/= { z } } )
2	1	elmpocl 6278	. . . . 5 |- ( x e. ( ( B X. { Z } ) supp Z ) -> ( ( B X. { Z } ) e. _V /\ Z e. _V ) )
3	2	simprd 114	. . . 4 |- ( x e. ( ( B X. { Z } ) supp Z ) -> Z e. _V )
4		fnconstg 5588	. . . . . . . 8 |- ( Z e. _V -> ( B X. { Z } ) Fn B )
5	3, 4	syl 14	. . . . . . 7 |- ( x e. ( ( B X. { Z } ) supp Z ) -> ( B X. { Z } ) Fn B )
6	2	simpld 112	. . . . . . 7 |- ( x e. ( ( B X. { Z } ) supp Z ) -> ( B X. { Z } ) e. _V )
7		elsuppfng 6476	. . . . . . 7 |- ( ( ( B X. { Z } ) Fn B /\ ( B X. { Z } ) e. _V /\ Z e. _V ) -> ( x e. ( ( B X. { Z } ) supp Z ) <-> ( x e. B /\ ( ( B X. { Z } ) ` x ) =/= Z ) ) )
8	5, 6, 3, 7	syl3anc 1278	. . . . . 6 |- ( x e. ( ( B X. { Z } ) supp Z ) -> ( x e. ( ( B X. { Z } ) supp Z ) <-> ( x e. B /\ ( ( B X. { Z } ) ` x ) =/= Z ) ) )
9	8	ibi 176	. . . . 5 |- ( x e. ( ( B X. { Z } ) supp Z ) -> ( x e. B /\ ( ( B X. { Z } ) ` x ) =/= Z ) )
10	9	simpld 112	. . . 4 |- ( x e. ( ( B X. { Z } ) supp Z ) -> x e. B )
11		fvconst2g 5923	. . . 4 |- ( ( Z e. _V /\ x e. B ) -> ( ( B X. { Z } ) ` x ) = Z )
12	3, 10, 11	syl2anc 415	. . 3 |- ( x e. ( ( B X. { Z } ) supp Z ) -> ( ( B X. { Z } ) ` x ) = Z )
13	9	simprd 114	. . . 4 |- ( x e. ( ( B X. { Z } ) supp Z ) -> ( ( B X. { Z } ) ` x ) =/= Z )
14	13	neneqd 2441	. . 3 |- ( x e. ( ( B X. { Z } ) supp Z ) -> -. ( ( B X. { Z } ) ` x ) = Z )
15	12, 14	pm2.65i 648	. 2 |- -. x e. ( ( B X. { Z } ) supp Z )
16	15	nel0 3543	1 |- ( ( B X. { Z } ) supp Z ) = (/)

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   {crab 2532   _Vcvv 2821   (/)c0 3520   {csn 3708   X. cxp 4770   dom cdm 4772   "cima 4775   Fn wfn 5370   ` cfv 5375  (class class class)co 6079   supp csupp 6469

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 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

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-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-supp 6470

This theorem is referenced by:  fczfsuppd  7291