Theorem qusin 17255

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 8581	. . . . 5 ⊢ ((( ∼ “ 𝑉) ⊆ 𝑉 ∧ 𝑥 ∈ 𝑉) → [𝑥] ∼ = [𝑥]( ∼ ∩ (𝑉 × 𝑉)))
3	1, 2	sylan 580	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → [𝑥] ∼ = [𝑥]( ∼ ∩ (𝑉 × 𝑉)))
4	3	mpteq2dva 5174	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) = (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))))
5	4	oveq1d 7290	. 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 2738	. . 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 17253	. 2 ⊢ (𝜑 → 𝑈 = ((𝑥 ∈ 𝑉 ↦ [𝑥] ∼ ) “s 𝑅))
12		eqidd 2739	. . 3 ⊢ (𝜑 → (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))) = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))))
13		eqid 2738	. . 3 ⊢ (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) = (𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉)))
14		inex1g 5243	. . . 4 ⊢ ( ∼ ∈ 𝑊 → ( ∼ ∩ (𝑉 × 𝑉)) ∈ V)
15	9, 14	syl 17	. . 3 ⊢ (𝜑 → ( ∼ ∩ (𝑉 × 𝑉)) ∈ V)
16	12, 7, 13, 15, 10	qusval 17253	. 2 ⊢ (𝜑 → (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))) = ((𝑥 ∈ 𝑉 ↦ [𝑥]( ∼ ∩ (𝑉 × 𝑉))) “s 𝑅))
17	5, 11, 16	3eqtr4d 2788	1 ⊢ (𝜑 → 𝑈 = (𝑅 /s ( ∼ ∩ (𝑉 × 𝑉))))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887   ↦ cmpt 5157   × cxp 5587   “ cima 5592  ‘cfv 6433  (class class class)co 7275  [cec 8496  Basecbs 16912   “_s cimas 17215   /_s cqus 17216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-ec 8500  df-qus 17220

This theorem is referenced by:  pi1addf  24210  pi1addval  24211  pi1grplem  24212