Theorem funisfsupp 7281

Description: The property of a function to be finitely supported (in relation to a given zero). (Contributed by AV, 23-May-2019.)

Assertion

Ref	Expression
funisfsupp	|- ( ( Fun R /\ R e. V /\ Z e. W ) -> ( R finSupp Z <-> ( R supp Z ) e. Fin ) )

Proof of Theorem funisfsupp

Step	Hyp	Ref	Expression
1		isfsupp 7279	. . 3 |- ( ( R e. V /\ Z e. W ) -> ( R finSupp Z <-> ( Fun R /\ ( R supp Z ) e. Fin ) ) )
2	1	3adant1 1046	. 2 |- ( ( Fun R /\ R e. V /\ Z e. W ) -> ( R finSupp Z <-> ( Fun R /\ ( R supp Z ) e. Fin ) ) )
3		ibar 301	. . . 4 |- ( Fun R -> ( ( R supp Z ) e. Fin <-> ( Fun R /\ ( R supp Z ) e. Fin ) ) )
4	3	bicomd 141	. . 3 |- ( Fun R -> ( ( Fun R /\ ( R supp Z ) e. Fin ) <-> ( R supp Z ) e. Fin ) )
5	4	3ad2ant1 1049	. 2 |- ( ( Fun R /\ R e. V /\ Z e. W ) -> ( ( Fun R /\ ( R supp Z ) e. Fin ) <-> ( R supp Z ) e. Fin ) )
6	2, 5	bitrd 188	1 |- ( ( Fun R /\ R e. V /\ Z e. W ) -> ( R finSupp Z <-> ( R supp Z ) e. Fin ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   e. wcel 2209   class class class wbr 4125   Fun wfun 5366  (class class class)co 6075   supp csupp 6465   Fincfn 7012   finSupp cfsupp 7275

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 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-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  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-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-rel 4776  df-cnv 4777  df-co 4778  df-iota 5332  df-fun 5374  df-fv 5380  df-ov 6078  df-fsupp 7276

This theorem is referenced by:  suppeqfsuppbi  7285  0fsupp  7288  ffsuppbi  7290  fcdmnn0fsuppg  9597