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

Theorem dpjval 19169
 Description: Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
Hypotheses
Ref Expression
dpjfval.1 (𝜑𝐺dom DProd 𝑆)
dpjfval.2 (𝜑 → dom 𝑆 = 𝐼)
dpjfval.p 𝑃 = (𝐺dProj𝑆)
dpjfval.q 𝑄 = (proj1𝐺)
dpjval.3 (𝜑𝑋𝐼)
Assertion
Ref Expression
dpjval (𝜑 → (𝑃𝑋) = ((𝑆𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))

Proof of Theorem dpjval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dpjfval.1 . . 3 (𝜑𝐺dom DProd 𝑆)
2 dpjfval.2 . . 3 (𝜑 → dom 𝑆 = 𝐼)
3 dpjfval.p . . 3 𝑃 = (𝐺dProj𝑆)
4 dpjfval.q . . 3 𝑄 = (proj1𝐺)
51, 2, 3, 4dpjfval 19168 . 2 (𝜑𝑃 = (𝑥𝐼 ↦ ((𝑆𝑥)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))))
6 simpr 488 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑥 = 𝑋)
76fveq2d 6656 . . 3 ((𝜑𝑥 = 𝑋) → (𝑆𝑥) = (𝑆𝑋))
86sneqd 4551 . . . . . 6 ((𝜑𝑥 = 𝑋) → {𝑥} = {𝑋})
98difeq2d 4074 . . . . 5 ((𝜑𝑥 = 𝑋) → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
109reseq2d 5831 . . . 4 ((𝜑𝑥 = 𝑋) → (𝑆 ↾ (𝐼 ∖ {𝑥})) = (𝑆 ↾ (𝐼 ∖ {𝑋})))
1110oveq2d 7156 . . 3 ((𝜑𝑥 = 𝑋) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))
127, 11oveq12d 7158 . 2 ((𝜑𝑥 = 𝑋) → ((𝑆𝑥)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) = ((𝑆𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
13 dpjval.3 . 2 (𝜑𝑋𝐼)
14 ovexd 7175 . 2 (𝜑 → ((𝑆𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) ∈ V)
155, 12, 13, 14fvmptd 6757 1 (𝜑 → (𝑃𝑋) = ((𝑆𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114  Vcvv 3469   ∖ cdif 3905  {csn 4539   class class class wbr 5042  dom cdm 5532   ↾ cres 5534  ‘cfv 6334  (class class class)co 7140  proj1cpj1 18751   DProd cdprd 19106  dProjcdpj 19107 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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-ixp 8449  df-dprd 19108  df-dpj 19109 This theorem is referenced by:  dpjf  19170  dpjidcl  19171  dpjlid  19174  dpjghm  19176
 Copyright terms: Public domain W3C validator