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Theorem isfsuppd 9366
Description: Deduction form of isfsupp 9365. (Contributed by SN, 29-Jul-2024.)
Hypotheses
Ref Expression
isfsuppd.r (𝜑𝑅𝑉)
isfsuppd.z (𝜑𝑍𝑊)
isfsuppd.1 (𝜑 → Fun 𝑅)
isfsuppd.2 (𝜑 → (𝑅 supp 𝑍) ∈ Fin)
Assertion
Ref Expression
isfsuppd (𝜑𝑅 finSupp 𝑍)

Proof of Theorem isfsuppd
StepHypRef Expression
1 isfsuppd.1 . 2 (𝜑 → Fun 𝑅)
2 isfsuppd.2 . 2 (𝜑 → (𝑅 supp 𝑍) ∈ Fin)
3 isfsuppd.r . . 3 (𝜑𝑅𝑉)
4 isfsuppd.z . . 3 (𝜑𝑍𝑊)
5 isfsupp 9365 . . 3 ((𝑅𝑉𝑍𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
63, 4, 5syl2anc 585 . 2 (𝜑 → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
71, 2, 6mpbir2and 712 1 (𝜑𝑅 finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107   class class class wbr 5149  Fun wfun 6538  (class class class)co 7409   supp csupp 8146  Fincfn 8939   finSupp cfsupp 9361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-rel 5684  df-cnv 5685  df-co 5686  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-fsupp 9362
This theorem is referenced by:  mptiffisupp  31915  elrspunsn  32547  selvvvval  41157  evlselvlem  41158  evlselv  41159
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