Theorem dpjval 18655

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 18654	. 2 ⊢ (𝜑 → 𝑃 = (𝑥 ∈ 𝐼 ↦ ((𝑆‘𝑥)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))))
6		simpr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
7	6	fveq2d 6356	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆‘𝑥) = (𝑆‘𝑋))
8	6	sneqd 4333	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → {𝑥} = {𝑋})
9	8	difeq2d 3871	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
10	9	reseq2d 5551	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝑆 ↾ (𝐼 ∖ {𝑥})) = (𝑆 ↾ (𝐼 ∖ {𝑋})))
11	10	oveq2d 6829	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))
12	7, 11	oveq12d 6831	. 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 6843	. 2 ⊢ (𝜑 → ((𝑆‘𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) ∈ V)
15	5, 12, 13, 14	fvmptd 6450	1 ⊢ (𝜑 → (𝑃‘𝑋) = ((𝑆‘𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ∖ cdif 3712  {csn 4321   class class class wbr 4804  dom cdm 5266   ↾ cres 5268  ‘cfv 6049  (class class class)co 6813  proj₁cpj1 18250   DProd cdprd 18592  dProjcdpj 18593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-ixp 8075  df-dprd 18594  df-dpj 18595

This theorem is referenced by:  dpjf  18656  dpjidcl  18657  dpjlid  18660  dpjghm  18662