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Mirrors > Home > MPE Home > Th. List > dprdwd | Structured version Visualization version GIF version |
Description: A mapping being a finitely supported function in the family 𝑆. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 11-Jul-2019.) (Proof shortened by OpenAI, 30-Mar-2020.) |
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) |
dprdwd.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ (𝑆‘𝑥)) |
dprdwd.4 | ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) finSupp 0 ) |
Ref | Expression |
---|---|
dprdwd | ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ 𝑊) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5033 | . . 3 ⊢ (ℎ = (𝑥 ∈ 𝐼 ↦ 𝐴) → (ℎ finSupp 0 ↔ (𝑥 ∈ 𝐼 ↦ 𝐴) finSupp 0 )) | |
2 | dprdwd.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ (𝑆‘𝑥)) | |
3 | 2 | ralrimiva 3149 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝐴 ∈ (𝑆‘𝑥)) |
4 | dprdff.1 | . . . . . . 7 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
5 | dprdff.2 | . . . . . . 7 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
6 | 4, 5 | dprddomcld 19116 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) |
7 | mptelixpg 8482 | . . . . . 6 ⊢ (𝐼 ∈ V → ((𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥) ↔ ∀𝑥 ∈ 𝐼 𝐴 ∈ (𝑆‘𝑥))) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥) ↔ ∀𝑥 ∈ 𝐼 𝐴 ∈ (𝑆‘𝑥))) |
9 | 3, 8 | mpbird 260 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥)) |
10 | fveq2 6645 | . . . . 5 ⊢ (𝑥 = 𝑖 → (𝑆‘𝑥) = (𝑆‘𝑖)) | |
11 | 10 | cbvixpv 8462 | . . . 4 ⊢ X𝑥 ∈ 𝐼 (𝑆‘𝑥) = X𝑖 ∈ 𝐼 (𝑆‘𝑖) |
12 | 9, 11 | eleqtrdi 2900 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖)) |
13 | dprdwd.4 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) finSupp 0 ) | |
14 | 1, 12, 13 | elrabd 3630 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 }) |
15 | dprdff.w | . 2 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } | |
16 | 14, 15 | eleqtrrdi 2901 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {crab 3110 Vcvv 3441 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 ‘cfv 6324 Xcixp 8444 finSupp cfsupp 8817 DProd cdprd 19108 |
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-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-oprab 7139 df-mpo 7140 df-ixp 8445 df-dprd 19110 |
This theorem is referenced by: dprdfid 19132 dprdfinv 19134 dprdfadd 19135 dmdprdsplitlem 19152 dpjidcl 19173 dchrptlem3 25850 |
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