Theorem fvn0elsuppb 6492

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 6491	. . . 4 |- ( ( ( B e. V /\ X e. B ) /\ ( G Fn B /\ ( G ` X ) =/= (/) ) ) -> X e. ( G supp (/) ) )
2	1	exp43 372	. . 3 |- ( B e. V -> ( X e. B -> ( G Fn B -> ( ( G ` X ) =/= (/) -> X e. ( G supp (/) ) ) ) ) )
3	2	3imp 1224	. 2 |- ( ( B e. V /\ X e. B /\ G Fn B ) -> ( ( G ` X ) =/= (/) -> X e. ( G supp (/) ) ) )
4		simp3 1030	. . . 4 |- ( ( B e. V /\ X e. B /\ G Fn B ) -> G Fn B )
5		simp1 1028	. . . 4 |- ( ( B e. V /\ X e. B /\ G Fn B ) -> B e. V )
6		0ex 4260	. . . . 5 |- (/) e. _V
7	6	a1i 9	. . . 4 |- ( ( B e. V /\ X e. B /\ G Fn B ) -> (/) e. _V )
8		elsuppfn 6483	. . . 4 |- ( ( G Fn B /\ B e. V /\ (/) e. _V ) -> ( X e. ( G supp (/) ) <-> ( X e. B /\ ( G ` X ) =/= (/) ) ) )
9	4, 5, 7, 8	syl3anc 1278	. . 3 |- ( ( B e. V /\ X e. B /\ G Fn B ) -> ( X e. ( G supp (/) ) <-> ( X e. B /\ ( G ` X ) =/= (/) ) ) )
10		simpr 110	. . 3 |- ( ( X e. B /\ ( G ` X ) =/= (/) ) -> ( G ` X ) =/= (/) )
11	9, 10	biimtrdi 163	. 2 |- ( ( B e. V /\ X e. B /\ G Fn B ) -> ( X e. ( G supp (/) ) -> ( G ` X ) =/= (/) ) )
12	3, 11	impbid 129	1 |- ( ( B e. V /\ X e. B /\ G Fn B ) -> ( ( G ` X ) =/= (/) <-> X e. ( G supp (/) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   e. wcel 2209   =/= wne 2420   _Vcvv 2821   (/)c0 3520   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   supp csupp 6475

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-supp 6476

This theorem is used by: (None)