Theorem isfsuppd 9368

Description: Deduction form of isfsupp 9367. (Contributed by SN, 29-Jul-2024.)

Hypotheses

Ref	Expression
isfsuppd.r	⊢ (𝜑 → 𝑅 ∈ 𝑉)
isfsuppd.z	⊢ (𝜑 → 𝑍 ∈ 𝑊)
isfsuppd.1	⊢ (𝜑 → Fun 𝑅)
isfsuppd.2	⊢ (𝜑 → (𝑅 supp 𝑍) ∈ Fin)

Assertion

Ref	Expression
isfsuppd	⊢ (𝜑 → 𝑅 finSupp 𝑍)

Proof of Theorem isfsuppd

Step	Hyp	Ref	Expression
1		isfsuppd.1	. 2 ⊢ (𝜑 → Fun 𝑅)
2		isfsuppd.2	. 2 ⊢ (𝜑 → (𝑅 supp 𝑍) ∈ Fin)
3		isfsuppd.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
4		isfsuppd.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑊)
5		isfsupp 9367	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
6	3, 4, 5	syl2anc 582	. 2 ⊢ (𝜑 → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin)))
7	1, 2, 6	mpbir2and 709	1 ⊢ (𝜑 → 𝑅 finSupp 𝑍)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∈ wcel 2104   class class class wbr 5147  Fun wfun 6536  (class class class)co 7411   supp csupp 8148  Fincfn 8941   finSupp cfsupp 9363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-rel 5682  df-cnv 5683  df-co 5684  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-fsupp 9364

This theorem is referenced by:  mptiffisupp  32182  elrspunsn  32821  selvvvval  41459  evlselvlem  41460  evlselv  41461