Theorem fczsupp0 6461

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 6438	. . . . . 6 |- supp = ( f e. _V , z e. _V |-> { q e. dom f | ( f " { q } ) =/= { z } } )
2	1	elmpocl 6251	. . . . 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 5567	. . . . . . . 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 6444	. . . . . . 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 1274	. . . . . 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 5900	. . . 4 |- ( ( Z e. _V /\ x e. B ) -> ( ( B X. { Z } ) ` x ) = Z )
12	3, 10, 11	syl2anc 411	. . 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 2435	. . 3 |- ( x e. ( ( B X. { Z } ) supp Z ) -> -. ( ( B X. { Z } ) ` x ) = Z )
15	12, 14	pm2.65i 644	. 2 |- -. x e. ( ( B X. { Z } ) supp Z )
16	15	nel0 3532	1 |- ( ( B X. { Z } ) supp Z ) = (/)

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   =/= wne 2414   {crab 2526   _Vcvv 2815   (/)c0 3510   {csn 3691   X. cxp 4749   dom cdm 4751   "cima 4754   Fn wfn 5349   ` cfv 5354  (class class class)co 6052   supp csupp 6437

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 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661

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 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-supp 6438

This theorem is referenced by:  fczfsuppd  7252