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Mirrors > Home > MPE Home > Th. List > Mathboxes > isfsuppd | Structured version Visualization version GIF version |
Description: Deduction form of isfsupp 8883. (Contributed by SN, 29-Jul-2024.) |
Ref | Expression |
---|---|
isfsuppd.r | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
isfsuppd.z | ⊢ (𝜑 → 𝑍 ∈ 𝑊) |
isfsuppd.1 | ⊢ (𝜑 → Fun 𝑅) |
isfsuppd.2 | ⊢ (𝜑 → (𝑅 supp 𝑍) ∈ Fin) |
Ref | Expression |
---|---|
isfsuppd | ⊢ (𝜑 → 𝑅 finSupp 𝑍) |
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 8883 | . . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))) | |
6 | 3, 4, 5 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝑅 finSupp 𝑍 ↔ (Fun 𝑅 ∧ (𝑅 supp 𝑍) ∈ Fin))) |
7 | 1, 2, 6 | mpbir2and 712 | 1 ⊢ (𝜑 → 𝑅 finSupp 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 class class class wbr 5036 Fun wfun 6334 (class class class)co 7156 supp csupp 7841 Fincfn 8540 finSupp cfsupp 8879 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pr 5302 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-sb 2070 df-clab 2736 df-cleq 2750 df-clel 2830 df-v 3411 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-br 5037 df-opab 5099 df-rel 5535 df-cnv 5536 df-co 5537 df-iota 6299 df-fun 6342 df-fv 6348 df-ov 7159 df-fsupp 8880 |
This theorem is referenced by: evlsbagval 39815 mhphf 39825 |
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