Theorem fsuppimp 7217

Description: Implications of a class being a finitely supported function (in relation to a given zero). (Contributed by AV, 26-May-2019.)

Assertion

Ref	Expression
fsuppimp	|- ( R finSupp Z -> ( Fun R /\ ( R supp Z ) e. Fin ) )

Proof of Theorem fsuppimp

Step	Hyp	Ref	Expression
1		relfsupp 7212	. . 3 |- Rel finSupp
2	1	brrelex12i 4774	. 2 |- ( R finSupp Z -> ( R e. _V /\ Z e. _V ) )
3		isfsupp 7214	. . 3 |- ( ( R e. _V /\ Z e. _V ) -> ( R finSupp Z <-> ( Fun R /\ ( R supp Z ) e. Fin ) ) )
4	3	biimpd 144	. 2 |- ( ( R e. _V /\ Z e. _V ) -> ( R finSupp Z -> ( Fun R /\ ( R supp Z ) e. Fin ) ) )
5	2, 4	mpcom 36	1 |- ( R finSupp Z -> ( Fun R /\ ( R supp Z ) e. Fin ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2202   _Vcvv 2803   class class class wbr 4093   Fun wfun 5327  (class class class)co 6028   supp csupp 6413   Fincfn 6952   finSupp cfsupp 7210

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-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-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  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-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-iota 5293  df-fun 5335  df-fv 5341  df-ov 6031  df-fsupp 7211

This theorem is referenced by:  fsuppimpd  7218  fsuppfund  7219