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Mirrors > Home > MPE Home > Th. List > dprdfcl | Structured version Visualization version GIF version |
Description: A finitely supported function in 𝑆 has its 𝑋-th element in 𝑆(𝑋). (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 |
---|---|
dprdfcl | ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐼) → (𝐹‘𝑋) ∈ (𝑆‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dprdff.3 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑊) | |
2 | dprdff.w | . . . . 5 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } | |
3 | dprdff.1 | . . . . 5 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
4 | dprdff.2 | . . . . 5 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
5 | 2, 3, 4 | dprdw 18853 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ 𝑊 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥) ∧ 𝐹 finSupp 0 ))) |
6 | 1, 5 | mpbid 233 | . . 3 ⊢ (𝜑 → (𝐹 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥) ∧ 𝐹 finSupp 0 )) |
7 | 6 | simp2d 1136 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥)) |
8 | fveq2 6545 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | |
9 | fveq2 6545 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑆‘𝑥) = (𝑆‘𝑋)) | |
10 | 8, 9 | eleq12d 2879 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ∈ (𝑆‘𝑥) ↔ (𝐹‘𝑋) ∈ (𝑆‘𝑋))) |
11 | 10 | rspccva 3560 | . 2 ⊢ ((∀𝑥 ∈ 𝐼 (𝐹‘𝑥) ∈ (𝑆‘𝑥) ∧ 𝑋 ∈ 𝐼) → (𝐹‘𝑋) ∈ (𝑆‘𝑋)) |
12 | 7, 11 | sylan 580 | 1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐼) → (𝐹‘𝑋) ∈ (𝑆‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ∀wral 3107 {crab 3111 class class class wbr 4968 dom cdm 5450 Fn wfn 6227 ‘cfv 6232 Xcixp 8317 finSupp cfsupp 8686 DProd cdprd 18836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-oprab 7027 df-mpo 7028 df-ixp 8318 df-dprd 18838 |
This theorem is referenced by: dprdfcntz 18858 dprdfinv 18862 dprdfadd 18863 dprdfeq0 18865 dprdlub 18869 dmdprdsplitlem 18880 dpjidcl 18901 |
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