Theorem qusin 17049

Description: Restrict the equivalence relation in a quotient structure to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
qusin.u	⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
qusin.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
qusin.e	⊢ (𝜑 → ∼ ∈ 𝑊)
qusin.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)
qusin.s	⊢ (𝜑 → ( ∼ “ 𝑉) ⊆ 𝑉)

Assertion

Ref	Expression
qusin	⊢ (𝜑 → 𝑈 = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))))

Proof of Theorem qusin

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusin.s	. . . . 5 ⊢ (𝜑 → ( ∼ “ 𝑉) ⊆ 𝑉)
2		ecinxp 8474	. . . . 5 ⊢ ((( ∼ “ 𝑉) ⊆ 𝑉 ∧ 𝑥 ∈ 𝑉) → [𝑥] ∼ = [𝑥]( ∼ ∩ (𝑉 × 𝑉)))
3	1, 2	sylan 583	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → [𝑥] ∼ = [𝑥]( ∼ ∩ (𝑉 × 𝑉)))
4	3	mpteq2dva 5150	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))))
5	4	oveq1d 7228	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) “s 𝑅) = ((𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) “s 𝑅))
6		qusin.u	. . 3 ⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
7		qusin.v	. . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
8		eqid 2737	. . 3 ⊢ (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
9		qusin.e	. . 3 ⊢ (𝜑 → ∼ ∈ 𝑊)
10		qusin.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
11	6, 7, 8, 9, 10	qusval 17047	. 2 ⊢ (𝜑 → 𝑈 = ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) “s 𝑅))
12		eqidd 2738	. . 3 ⊢ (𝜑 → (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))) = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))))
13		eqid 2737	. . 3 ⊢ (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) = (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉)))
14		inex1g 5212	. . . 4 ⊢ ( ∼ ∈ 𝑊 → ( ∼ ∩ (𝑉 × 𝑉)) ∈ V)
15	9, 14	syl 17	. . 3 ⊢ (𝜑 → ( ∼ ∩ (𝑉 × 𝑉)) ∈ V)
16	12, 7, 13, 15, 10	qusval 17047	. 2 ⊢ (𝜑 → (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))) = ((𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) “s 𝑅))
17	5, 11, 16	3eqtr4d 2787	1 ⊢ (𝜑 → 𝑈 = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))))

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2110  Vcvv 3408   ∩ cin 3865   ⊆ wss 3866   ↦ cmpt 5135   × cxp 5549   “ cima 5554  ‘cfv 6380  (class class class)co 7213  [cec 8389  Basecbs 16760   “_s cimas 17009   /_s cqus 17010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-ec 8393  df-qus 17014

This theorem is referenced by:  pi1addf  23944  pi1addval  23945  pi1grplem  23946