Theorem fvn0elsupp 6429

Description: If the function value for a given argument is not empty, the argument belongs to the support of the function with the empty set as zero. (Contributed by AV, 2-Jul-2019.) (Revised by AV, 4-Apr-2020.)

Assertion

Ref	Expression
fvn0elsupp	|- ( ( ( B e. V /\ X e. B ) /\ ( G Fn B /\ ( G ` X ) =/= (/) ) ) -> X e. ( G supp (/) ) )

Proof of Theorem fvn0elsupp

Step	Hyp	Ref	Expression
1		simpr 110	. . 3 |- ( ( B e. V /\ X e. B ) -> X e. B )
2		simpr 110	. . 3 |- ( ( G Fn B /\ ( G ` X ) =/= (/) ) -> ( G ` X ) =/= (/) )
3	1, 2	anim12i 338	. 2 |- ( ( ( B e. V /\ X e. B ) /\ ( G Fn B /\ ( G ` X ) =/= (/) ) ) -> ( X e. B /\ ( G ` X ) =/= (/) ) )
4		simprl 531	. . 3 |- ( ( ( B e. V /\ X e. B ) /\ ( G Fn B /\ ( G ` X ) =/= (/) ) ) -> G Fn B )
5		simpll 527	. . 3 |- ( ( ( B e. V /\ X e. B ) /\ ( G Fn B /\ ( G ` X ) =/= (/) ) ) -> B e. V )
6		0ex 4221	. . . 4 |- (/) e. _V
7	6	a1i 9	. . 3 |- ( ( ( B e. V /\ X e. B ) /\ ( G Fn B /\ ( G ` X ) =/= (/) ) ) -> (/) e. _V )
8		elsuppfn 6421	. . 3 |- ( ( G Fn B /\ B e. V /\ (/) e. _V ) -> ( X e. ( G supp (/) ) <-> ( X e. B /\ ( G ` X ) =/= (/) ) ) )
9	4, 5, 7, 8	syl3anc 1274	. 2 |- ( ( ( B e. V /\ X e. B ) /\ ( G Fn B /\ ( G ` X ) =/= (/) ) ) -> ( X e. ( G supp (/) ) <-> ( X e. B /\ ( G ` X ) =/= (/) ) ) )
10	3, 9	mpbird 167	1 |- ( ( ( B e. V /\ X e. B ) /\ ( G Fn B /\ ( G ` X ) =/= (/) ) ) -> X e. ( G supp (/) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2202   =/= wne 2403   _Vcvv 2803   (/)c0 3496   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   supp csupp 6413

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-supp 6414

This theorem is referenced by:  fvn0elsuppb  6430