Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dprdfcl Structured version   Visualization version   GIF version

Theorem dprdfcl 18458
 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 18455 . . . 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 1094 . 2 (𝜑 → ∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥))
8 fveq2 6229 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
9 fveq2 6229 . . . 4 (𝑥 = 𝑋 → (𝑆𝑥) = (𝑆𝑋))
108, 9eleq12d 2724 . . 3 (𝑥 = 𝑋 → ((𝐹𝑥) ∈ (𝑆𝑥) ↔ (𝐹𝑋) ∈ (𝑆𝑋)))
1110rspccva 3339 . 2 ((∀𝑥𝐼 (𝐹𝑥) ∈ (𝑆𝑥) ∧ 𝑋𝐼) → (𝐹𝑋) ∈ (𝑆𝑋))
127, 11sylan 487 1 ((𝜑𝑋𝐼) → (𝐹𝑋) ∈ (𝑆𝑋))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  {crab 2945   class class class wbr 4685  dom cdm 5143   Fn wfn 5921  ‘cfv 5926  Xcixp 7950   finSupp cfsupp 8316   DProd cdprd 18438 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-oprab 6694  df-mpt2 6695  df-ixp 7951  df-dprd 18440 This theorem is referenced by:  dprdfcntz  18460  dprdfinv  18464  dprdfadd  18465  dprdfeq0  18467  dprdlub  18471  dmdprdsplitlem  18482  dpjidcl  18503
 Copyright terms: Public domain W3C validator