Theorem isfsuppd 7243

Description: Deduction form of isfsupp 7242. (Contributed by SN, 29-Jul-2024.)

Hypotheses

Ref	Expression
isfsuppd.r	|- ( ph -> R e. V )
isfsuppd.z	|- ( ph -> Z e. W )
isfsuppd.1	|- ( ph -> Fun R )
isfsuppd.2	|- ( ph -> ( R supp Z ) e. Fin )

Assertion

Ref	Expression
isfsuppd	|- ( ph -> R finSupp Z )

Proof of Theorem isfsuppd

Step	Hyp	Ref	Expression
1		isfsuppd.1	. 2 |- ( ph -> Fun R )
2		isfsuppd.2	. 2 |- ( ph -> ( R supp Z ) e. Fin )
3		isfsuppd.r	. . 3 |- ( ph -> R e. V )
4		isfsuppd.z	. . 3 |- ( ph -> Z e. W )
5		isfsupp 7242	. . 3 |- ( ( R e. V /\ Z e. W ) -> ( R finSupp Z <-> ( Fun R /\ ( R supp Z ) e. Fin ) ) )
6	3, 4, 5	syl2anc 411	. 2 |- ( ph -> ( R finSupp Z <-> ( Fun R /\ ( R supp Z ) e. Fin ) ) )
7	1, 2, 6	mpbir2and 953	1 |- ( ph -> R finSupp Z )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2203   class class class wbr 4109   Fun wfun 5346  (class class class)co 6050   supp csupp 6435   Fincfn 6975   finSupp cfsupp 7238

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-rel 4756  df-cnv 4757  df-co 4758  df-iota 5312  df-fun 5354  df-fv 5360  df-ov 6053  df-fsupp 7239

This theorem is referenced by:  fczfsuppd  7250  snopfsuppdc  7252  fsuppcorn  7254