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Theorem dpjval 19925
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 19924 . 2 (𝜑𝑃 = (𝑥𝐼 ↦ ((𝑆𝑥)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))))
6 simpr 485 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑥 = 𝑋)
76fveq2d 6895 . . 3 ((𝜑𝑥 = 𝑋) → (𝑆𝑥) = (𝑆𝑋))
86sneqd 4640 . . . . . 6 ((𝜑𝑥 = 𝑋) → {𝑥} = {𝑋})
98difeq2d 4122 . . . . 5 ((𝜑𝑥 = 𝑋) → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
109reseq2d 5981 . . . 4 ((𝜑𝑥 = 𝑋) → (𝑆 ↾ (𝐼 ∖ {𝑥})) = (𝑆 ↾ (𝐼 ∖ {𝑋})))
1110oveq2d 7424 . . 3 ((𝜑𝑥 = 𝑋) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))
127, 11oveq12d 7426 . 2 ((𝜑𝑥 = 𝑋) → ((𝑆𝑥)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) = ((𝑆𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
13 dpjval.3 . 2 (𝜑𝑋𝐼)
14 ovexd 7443 . 2 (𝜑 → ((𝑆𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) ∈ V)
155, 12, 13, 14fvmptd 7005 1 (𝜑 → (𝑃𝑋) = ((𝑆𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  cdif 3945  {csn 4628   class class class wbr 5148  dom cdm 5676  cres 5678  cfv 6543  (class class class)co 7408  proj1cpj1 19502   DProd cdprd 19862  dProjcdpj 19863
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-ixp 8891  df-dprd 19864  df-dpj 19865
This theorem is referenced by:  dpjf  19926  dpjidcl  19927  dpjlid  19930  dpjghm  19932
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