Theorem dpjval 19963

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 19962	. 2 ⊢ (𝜑 → 𝑃 = (𝑥 ∈ 𝐼 ↦ ((𝑆‘𝑥)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))))
6		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
7	6	fveq2d 6821	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆‘𝑥) = (𝑆‘𝑋))
8	6	sneqd 4586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → {𝑥} = {𝑋})
9	8	difeq2d 4074	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
10	9	reseq2d 5925	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆 ↾ (𝐼 ∖ {𝑥})) = (𝑆 ↾ (𝐼 ∖ {𝑋})))
11	10	oveq2d 7357	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))
12	7, 11	oveq12d 7359	. 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 7376	. 2 ⊢ (𝜑 → ((𝑆‘𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) ∈ V)
15	5, 12, 13, 14	fvmptd 6931	1 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110  Vcvv 3434   ∖ cdif 3897  {csn 4574   class class class wbr 5089  dom cdm 5614   ↾ cres 5616  ‘cfv 6477  (class class class)co 7341  proj₁cpj1 19540   DProd cdprd 19900  dProjcdpj 19901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-ixp 8817  df-dprd 19902  df-dpj 19903

This theorem is referenced by:  dpjf  19964  dpjidcl  19965  dpjlid  19968  dpjghm  19970