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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 5072 | . . 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 3185 | . . . . 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 19126 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ V) |
7 | mptelixpg 8502 | . . . . . 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 259 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑥 ∈ 𝐼 (𝑆‘𝑥)) |
10 | fveq2 6673 | . . . . 5 ⊢ (𝑥 = 𝑖 → (𝑆‘𝑥) = (𝑆‘𝑖)) | |
11 | 10 | cbvixpv 8482 | . . . 4 ⊢ X𝑥 ∈ 𝐼 (𝑆‘𝑥) = X𝑖 ∈ 𝐼 (𝑆‘𝑖) |
12 | 9, 11 | eleqtrdi 2926 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖)) |
13 | dprdwd.4 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) finSupp 0 ) | |
14 | 1, 12, 13 | elrabd 3685 | . 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 2927 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ 𝐴) ∈ 𝑊) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 {crab 3145 Vcvv 3497 class class class wbr 5069 ↦ cmpt 5149 dom cdm 5558 ‘cfv 6358 Xcixp 8464 finSupp cfsupp 8836 DProd cdprd 19118 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-oprab 7163 df-mpo 7164 df-ixp 8465 df-dprd 19120 |
This theorem is referenced by: dprdfid 19142 dprdfinv 19144 dprdfadd 19145 dmdprdsplitlem 19162 dpjidcl 19183 dchrptlem3 25845 |
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