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Theorem dpjval 19926
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 19925 . 2 (𝜑𝑃 = (𝑥𝐼 ↦ ((𝑆𝑥)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))))
6 simpr 486 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑥 = 𝑋)
76fveq2d 6896 . . 3 ((𝜑𝑥 = 𝑋) → (𝑆𝑥) = (𝑆𝑋))
86sneqd 4641 . . . . . 6 ((𝜑𝑥 = 𝑋) → {𝑥} = {𝑋})
98difeq2d 4123 . . . . 5 ((𝜑𝑥 = 𝑋) → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
109reseq2d 5982 . . . 4 ((𝜑𝑥 = 𝑋) → (𝑆 ↾ (𝐼 ∖ {𝑥})) = (𝑆 ↾ (𝐼 ∖ {𝑋})))
1110oveq2d 7425 . . 3 ((𝜑𝑥 = 𝑋) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))
127, 11oveq12d 7427 . 2 ((𝜑𝑥 = 𝑋) → ((𝑆𝑥)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) = ((𝑆𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
13 dpjval.3 . 2 (𝜑𝑋𝐼)
14 ovexd 7444 . 2 (𝜑 → ((𝑆𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) ∈ V)
155, 12, 13, 14fvmptd 7006 1 (𝜑 → (𝑃𝑋) = ((𝑆𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  Vcvv 3475  cdif 3946  {csn 4629   class class class wbr 5149  dom cdm 5677  cres 5679  cfv 6544  (class class class)co 7409  proj1cpj1 19503   DProd cdprd 19863  dProjcdpj 19864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-ixp 8892  df-dprd 19865  df-dpj 19866
This theorem is referenced by:  dpjf  19927  dpjidcl  19928  dpjlid  19931  dpjghm  19933
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