Theorem fczsupp0 6458

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 6435	. . . . . 6 |- supp = ( f e. _V , z e. _V |-> { q e. dom f | ( f " { q } ) =/= { z } } )
2	1	elmpocl 6248	. . . . 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 5564	. . . . . . . 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 6441	. . . . . . 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 5897	. . . 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 2433	. . 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 3529	1 |- ( ( B X. { Z } ) supp Z ) = (/)

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   =/= wne 2412   {crab 2524   _Vcvv 2812   (/)c0 3507   {csn 3688   X. cxp 4746   dom cdm 4748   "cima 4751   Fn wfn 5346   ` cfv 5351  (class class class)co 6049   supp csupp 6434

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-supp 6435

This theorem is referenced by:  fczfsuppd  7249