Theorem fvn0elsuppb 7312

Description: The function value for a given argument is not empty iff the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 4-Apr-2020.)

Assertion

Ref	Expression
fvn0elsuppb	|- ( ( B e. V /\ X e. B /\ G Fn B ) -> ( ( G ` X ) =/= (/) <-> X e. ( G supp (/) ) ) )

Proof of Theorem fvn0elsuppb

Step	Hyp	Ref	Expression
1		fvn0elsupp 7311	. . . 4 |- ( ( ( B e. V /\ X e. B ) /\ ( G Fn B /\ ( G ` X ) =/= (/) ) ) -> X e. ( G supp (/) ) )
2	1	exp43 640	. . 3 |- ( B e. V -> ( X e. B -> ( G Fn B -> ( ( G ` X ) =/= (/) -> X e. ( G supp (/) ) ) ) ) )
3	2	3imp 1256	. 2 |- ( ( B e. V /\ X e. B /\ G Fn B ) -> ( ( G ` X ) =/= (/) -> X e. ( G supp (/) ) ) )
4		simp3 1063	. . . 4 |- ( ( B e. V /\ X e. B /\ G Fn B ) -> G Fn B )
5		simp1 1061	. . . 4 |- ( ( B e. V /\ X e. B /\ G Fn B ) -> B e. V )
6		0ex 4790	. . . . 5 |- (/) e. _V
7	6	a1i 11	. . . 4 |- ( ( B e. V /\ X e. B /\ G Fn B ) -> (/) e. _V )
8		elsuppfn 7303	. . . 4 |- ( ( G Fn B /\ B e. V /\ (/) e. _V ) -> ( X e. ( G supp (/) ) <-> ( X e. B /\ ( G ` X ) =/= (/) ) ) )
9	4, 5, 7, 8	syl3anc 1326	. . 3 |- ( ( B e. V /\ X e. B /\ G Fn B ) -> ( X e. ( G supp (/) ) <-> ( X e. B /\ ( G ` X ) =/= (/) ) ) )
10		simpr 477	. . 3 |- ( ( X e. B /\ ( G ` X ) =/= (/) ) -> ( G ` X ) =/= (/) )
11	9, 10	syl6bi 243	. 2 |- ( ( B e. V /\ X e. B /\ G Fn B ) -> ( X e. ( G supp (/) ) -> ( G ` X ) =/= (/) ) )
12	3, 11	impbid 202	1 |- ( ( B e. V /\ X e. B /\ G Fn B ) -> ( ( G ` X ) =/= (/) <-> X e. ( G supp (/) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   Fn wfn 5883   ` cfv 5888  (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-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:  brcic  16458