Theorem qusin 17622

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 8796	. . . . 5 ⊢ ((( ∼ “ 𝑉) ⊆ 𝑉 ∧ 𝑥 ∈ 𝑉) → [𝑥] ∼ = [𝑥]( ∼ ∩ (𝑉 × 𝑉)))
3	1, 2	sylan 592	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → [𝑥] ∼ = [𝑥]( ∼ ∩ (𝑉 × 𝑉)))
4	3	mpteq2dva 5206	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))))
5	4	oveq1d 7434	. 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 2765	. . 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 17620	. 2 ⊢ (𝜑 → 𝑈 = ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) “s 𝑅))
12		eqidd 2766	. . 3 ⊢ (𝜑 → (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))) = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))))
13		eqid 2765	. . 3 ⊢ (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) = (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉)))
14		inex1g 5290	. . . 4 ⊢ ( ∼ ∈ 𝑊 → ( ∼ ∩ (𝑉 × 𝑉)) ∈ V)
15	9, 14	syl 18	. . 3 ⊢ (𝜑 → ( ∼ ∩ (𝑉 × 𝑉)) ∈ V)
16	12, 7, 13, 15, 10	qusval 17620	. 2 ⊢ (𝜑 → (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))) = ((𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) “s 𝑅))
17	5, 11, 16	3eqtr4d 2810	1 ⊢ (𝜑 → 𝑈 = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ∩ cin 3905   ⊆ wss 3906   ↦ cmpt 5194   × cxp 5661   “ cima 5666  ‘cfv 6540  (class class class)co 7419  [cec 8698  Basecbs 17293   “_s cimas 17582   /_s cqus 17583

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-ec 8702  df-qus 17587

This theorem is used by:  pi1addf  25259  pi1addval  25260  pi1grplem  25261