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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
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