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Mirrors > Home > MPE Home > Th. List > dprdffsupp | Structured version Visualization version GIF version |
Description: A finitely supported function in 𝑆 is a finitely supported function. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.) |
Ref | Expression |
---|---|
dprdff.w | ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } |
dprdff.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
dprdff.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
dprdff.3 | ⊢ (𝜑 → 𝐹 ∈ 𝑊) |
Ref | Expression |
---|---|
dprdffsupp | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dprdff.3 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊) | |
2 | dprdff.w | . . . 4 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } | |
3 | dprdff.1 | . . . 4 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
4 | dprdff.2 | . . . 4 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
5 | 2, 3, 4 | dprdw 19126 | . . 3 ⊢ (𝜑 → (𝐹 ∈ 𝑊 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥) ∧ 𝐹 finSupp 0 ))) |
6 | 1, 5 | mpbid 234 | . 2 ⊢ (𝜑 → (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥) ∧ 𝐹 finSupp 0 )) |
7 | 6 | simp3d 1140 | 1 ⊢ (𝜑 → 𝐹 finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {crab 3142 class class class wbr 5058 dom cdm 5549 Fn wfn 6344 ‘cfv 6349 Xcixp 8455 finSupp cfsupp 8827 DProd cdprd 19109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-oprab 7154 df-mpo 7155 df-ixp 8456 df-dprd 19111 |
This theorem is referenced by: dprdssv 19132 dprdfinv 19135 dprdfadd 19136 dprdfeq0 19138 dprdlub 19142 dmdprdsplitlem 19153 dpjidcl 19174 |
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