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Theorem dprdfcl 18856
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 18853 . . . 4 (𝜑 → (𝐹𝑊 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥) ∧ 𝐹 finSupp 0 )))
61, 5mpbid 233 . . 3 (𝜑 → (𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥) ∧ 𝐹 finSupp 0 ))
76simp2d 1136 . 2 (𝜑 → ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥))
8 fveq2 6545 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
9 fveq2 6545 . . . 4 (𝑥 = 𝑋 → (𝑆𝑥) = (𝑆𝑋))
108, 9eleq12d 2879 . . 3 (𝑥 = 𝑋 → ((𝐹𝑥) ∈ (𝑆𝑥) ↔ (𝐹𝑋) ∈ (𝑆𝑋)))
1110rspccva 3560 . 2 ((∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥) ∧ 𝑋𝐼) → (𝐹𝑋) ∈ (𝑆𝑋))
127, 11sylan 580 1 ((𝜑𝑋𝐼) → (𝐹𝑋) ∈ (𝑆𝑋))
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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