Theorem dpjval 20017

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	⊢ (𝜑 → 𝐺dom DProd 𝑆)
dpjfval.2	⊢ (𝜑 → dom 𝑆 = 𝐼)
dpjfval.p	⊢ 𝑃 = (𝐺dProj𝑆)
dpjfval.q	⊢ 𝑄 = (proj1‘𝐺)
dpjval.3	⊢ (𝜑 → 𝑋 ∈ 𝐼)

Assertion

Ref	Expression
dpjval	⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))

Proof of Theorem dpjval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dpjfval.1	. . 3 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
2		dpjfval.2	. . 3 ⊢ (𝜑 → dom 𝑆 = 𝐼)
3		dpjfval.p	. . 3 ⊢ 𝑃 = (𝐺dProj𝑆)
4		dpjfval.q	. . 3 ⊢ 𝑄 = (proj1‘𝐺)
5	1, 2, 3, 4	dpjfval 20016	. 2 ⊢ (𝜑 → 𝑃 = (𝑥 ∈ 𝐼 ↦ ((𝑆‘𝑥)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))))
6		simpr 483	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
7	6	fveq2d 6898	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆‘𝑥) = (𝑆‘𝑋))
8	6	sneqd 4641	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → {𝑥} = {𝑋})
9	8	difeq2d 4119	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
10	9	reseq2d 5984	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆 ↾ (𝐼 ∖ {𝑥})) = (𝑆 ↾ (𝐼 ∖ {𝑋})))
11	10	oveq2d 7433	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))
12	7, 11	oveq12d 7435	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝑆‘𝑥)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) = ((𝑆‘𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
13		dpjval.3	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
14		ovexd 7452	. 2 ⊢ (𝜑 → ((𝑆‘𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) ∈ V)
15	5, 12, 13, 14	fvmptd 7009	1 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463   ∖ cdif 3942  {csn 4629   class class class wbr 5148  dom cdm 5677   ↾ cres 5679  ‘cfv 6547  (class class class)co 7417  proj₁cpj1 19594   DProd cdprd 19954  dProjcdpj 19955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  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 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7992  df-2nd 7993  df-ixp 8915  df-dprd 19956  df-dpj 19957

This theorem is referenced by:  dpjf  20018  dpjidcl  20019  dpjlid  20022  dpjghm  20024