Theorem dpjval 19976

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 19975	. 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 6832	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆‘𝑥) = (𝑆‘𝑋))
8	6	sneqd 4587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → {𝑥} = {𝑋})
9	8	difeq2d 4075	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
10	9	reseq2d 5933	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆 ↾ (𝐼 ∖ {𝑥})) = (𝑆 ↾ (𝐼 ∖ {𝑋})))
11	10	oveq2d 7368	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))
12	7, 11	oveq12d 7370	. 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 7387	. 2 ⊢ (𝜑 → ((𝑆‘𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) ∈ V)
15	5, 12, 13, 14	fvmptd 6942	1 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436   ∖ cdif 3894  {csn 4575   class class class wbr 5093  dom cdm 5619   ↾ cres 5621  ‘cfv 6487  (class class class)co 7352  proj₁cpj1 19553   DProd cdprd 19913  dProjcdpj 19914

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

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 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-ixp 8828  df-dprd 19915  df-dpj 19916

This theorem is referenced by:  dpjf  19977  dpjidcl  19978  dpjlid  19981  dpjghm  19983