Theorem fnsuppeq0 7323

Description: The support of a function is empty iff it is identically zero. (Contributed by Stefan O'Rear, 22-Mar-2015.) (Revised by AV, 28-May-2019.)

Assertion

Ref	Expression
fnsuppeq0	|- ( ( F Fn A /\ A e. W /\ Z e. V ) -> ( ( F supp Z ) = (/) <-> F = ( A X. { Z } ) ) )

Proof of Theorem fnsuppeq0

Step	Hyp	Ref	Expression
1		ss0b 3973	. . 3 |- ( ( F supp Z ) C_ (/) <-> ( F supp Z ) = (/) )
2		un0 3967	. . . . . . . 8 |- ( A u. (/) ) = A
3		uncom 3757	. . . . . . . 8 |- ( A u. (/) ) = ( (/) u. A )
4	2, 3	eqtr3i 2646	. . . . . . 7 |- A = ( (/) u. A )
5	4	fneq2i 5986	. . . . . 6 |- ( F Fn A <-> F Fn ( (/) u. A ) )
6	5	biimpi 206	. . . . 5 |- ( F Fn A -> F Fn ( (/) u. A ) )
7	6	3ad2ant1 1082	. . . 4 |- ( ( F Fn A /\ A e. W /\ Z e. V ) -> F Fn ( (/) u. A ) )
8		fnex 6481	. . . . 5 |- ( ( F Fn A /\ A e. W ) -> F e. _V )
9	8	3adant3 1081	. . . 4 |- ( ( F Fn A /\ A e. W /\ Z e. V ) -> F e. _V )
10		simp3 1063	. . . 4 |- ( ( F Fn A /\ A e. W /\ Z e. V ) -> Z e. V )
11		0in 3969	. . . . 5 |- ( (/) i^i A ) = (/)
12	11	a1i 11	. . . 4 |- ( ( F Fn A /\ A e. W /\ Z e. V ) -> ( (/) i^i A ) = (/) )
13		fnsuppres 7322	. . . 4 |- ( ( F Fn ( (/) u. A ) /\ ( F e. _V /\ Z e. V ) /\ ( (/) i^i A ) = (/) ) -> ( ( F supp Z ) C_ (/) <-> ( F |` A ) = ( A X. { Z } ) ) )
14	7, 9, 10, 12, 13	syl121anc 1331	. . 3 |- ( ( F Fn A /\ A e. W /\ Z e. V ) -> ( ( F supp Z ) C_ (/) <-> ( F |` A ) = ( A X. { Z } ) ) )
15	1, 14	syl5bbr 274	. 2 |- ( ( F Fn A /\ A e. W /\ Z e. V ) -> ( ( F supp Z ) = (/) <-> ( F |` A ) = ( A X. { Z } ) ) )
16		fnresdm 6000	. . . 4 |- ( F Fn A -> ( F |` A ) = F )
17	16	3ad2ant1 1082	. . 3 |- ( ( F Fn A /\ A e. W /\ Z e. V ) -> ( F |` A ) = F )
18	17	eqeq1d 2624	. 2 |- ( ( F Fn A /\ A e. W /\ Z e. V ) -> ( ( F |` A ) = ( A X. { Z } ) <-> F = ( A X. { Z } ) ) )
19	15, 18	bitrd 268	1 |- ( ( F Fn A /\ A e. W /\ Z e. V ) -> ( ( F supp Z ) = (/) <-> F = ( A X. { Z } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   X. cxp 5112   |` cres 5116   Fn wfn 5883  (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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-supp 7296

This theorem is referenced by:  fczsupp0  7324  cantnf0  8572  mdegldg  23826  mdeg0  23830