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Theorem dprdfcl 18352
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.)
Hypotheses
Ref Expression
dprdff.w 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
dprdff.1 (𝜑𝐺dom DProd 𝑆)
dprdff.2 (𝜑 → dom 𝑆 = 𝐼)
dprdff.3 (𝜑𝐹𝑊)
Assertion
Ref Expression
dprdfcl ((𝜑𝑋𝐼) → (𝐹𝑋) ∈ (𝑆𝑋))
Distinct variable groups:   ,𝐹   ,𝑖,𝐼   0 ,   𝑆,,𝑖
Allowed substitution hints:   𝜑(,𝑖)   𝐹(𝑖)   𝐺(,𝑖)   𝑊(,𝑖)   𝑋(,𝑖)   0 (𝑖)

Proof of Theorem dprdfcl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dprdff.3 . . . 4 (𝜑𝐹𝑊)
2 dprdff.w . . . . 5 𝑊 = {X𝑖𝐼 (𝑆𝑖) ∣ finSupp 0 }
3 dprdff.1 . . . . 5 (𝜑𝐺dom DProd 𝑆)
4 dprdff.2 . . . . 5 (𝜑 → dom 𝑆 = 𝐼)
52, 3, 4dprdw 18349 . . . 4 (𝜑 → (𝐹𝑊 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥) ∧ 𝐹 finSupp 0 )))
61, 5mpbid 222 . . 3 (𝜑 → (𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥) ∧ 𝐹 finSupp 0 ))
76simp2d 1072 . 2 (𝜑 → ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥))
8 fveq2 6158 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
9 fveq2 6158 . . . 4 (𝑥 = 𝑋 → (𝑆𝑥) = (𝑆𝑋))
108, 9eleq12d 2692 . . 3 (𝑥 = 𝑋 → ((𝐹𝑥) ∈ (𝑆𝑥) ↔ (𝐹𝑋) ∈ (𝑆𝑋)))
1110rspccva 3298 . 2 ((∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥) ∧ 𝑋𝐼) → (𝐹𝑋) ∈ (𝑆𝑋))
127, 11sylan 488 1 ((𝜑𝑋𝐼) → (𝐹𝑋) ∈ (𝑆𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  {crab 2912   class class class wbr 4623  dom cdm 5084   Fn wfn 5852  cfv 5857  Xcixp 7868   finSupp cfsupp 8235   DProd cdprd 18332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-oprab 6619  df-mpt2 6620  df-ixp 7869  df-dprd 18334
This theorem is referenced by:  dprdfcntz  18354  dprdfinv  18358  dprdfadd  18359  dprdfeq0  18361  dprdlub  18365  dmdprdsplitlem  18376  dpjidcl  18397
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