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Theorem dprdfcl 19203
 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 19200 . . . 4 (𝜑 → (𝐹𝑊 ↔ (𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥) ∧ 𝐹 finSupp 0 )))
61, 5mpbid 235 . . 3 (𝜑 → (𝐹 Fn 𝐼 ∧ ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥) ∧ 𝐹 finSupp 0 ))
76simp2d 1140 . 2 (𝜑 → ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥))
8 fveq2 6658 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
9 fveq2 6658 . . . 4 (𝑥 = 𝑋 → (𝑆𝑥) = (𝑆𝑋))
108, 9eleq12d 2846 . . 3 (𝑥 = 𝑋 → ((𝐹𝑥) ∈ (𝑆𝑥) ↔ (𝐹𝑋) ∈ (𝑆𝑋)))
1110rspccva 3540 . 2 ((∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥) ∧ 𝑋𝐼) → (𝐹𝑋) ∈ (𝑆𝑋))
127, 11sylan 583 1 ((𝜑𝑋𝐼) → (𝐹𝑋) ∈ (𝑆𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  {crab 3074   class class class wbr 5032  dom cdm 5524   Fn wfn 6330  ‘cfv 6335  Xcixp 8479   finSupp cfsupp 8866   DProd cdprd 19183 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 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-oprab 7154  df-mpo 7155  df-ixp 8480  df-dprd 19185 This theorem is referenced by:  dprdfcntz  19205  dprdfinv  19209  dprdfadd  19210  dprdfeq0  19212  dprdlub  19216  dmdprdsplitlem  19227  dpjidcl  19248
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