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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 (𝜑𝑈 = (𝑅 /s ))
qusin.v (𝜑𝑉 = (Base‘𝑅))
qusin.e (𝜑𝑊)
qusin.r (𝜑𝑅𝑍)
qusin.s (𝜑 → ( 𝑉) ⊆ 𝑉)
Assertion
Ref Expression
qusin (𝜑𝑈 = (𝑅 /s ( ∩ (𝑉 × 𝑉))))

Proof of Theorem qusin
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 qusin.s . . . . 5 (𝜑 → ( 𝑉) ⊆ 𝑉)
2 ecinxp 8474 . . . . 5 ((( 𝑉) ⊆ 𝑉𝑥𝑉) → [𝑥] = [𝑥]( ∩ (𝑉 × 𝑉)))
31, 2sylan 583 . . . 4 ((𝜑𝑥𝑉) → [𝑥] = [𝑥]( ∩ (𝑉 × 𝑉)))
43mpteq2dva 5150 . . 3 (𝜑 → (𝑥𝑉 ↦ [𝑥] ) = (𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉))))
54oveq1d 7228 . 2 (𝜑 → ((𝑥𝑉 ↦ [𝑥] ) “s 𝑅) = ((𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉))) “s 𝑅))
6 qusin.u . . 3 (𝜑𝑈 = (𝑅 /s ))
7 qusin.v . . 3 (𝜑𝑉 = (Base‘𝑅))
8 eqid 2737 . . 3 (𝑥𝑉 ↦ [𝑥] ) = (𝑥𝑉 ↦ [𝑥] )
9 qusin.e . . 3 (𝜑𝑊)
10 qusin.r . . 3 (𝜑𝑅𝑍)
116, 7, 8, 9, 10qusval 17047 . 2 (𝜑𝑈 = ((𝑥𝑉 ↦ [𝑥] ) “s 𝑅))
12 eqidd 2738 . . 3 (𝜑 → (𝑅 /s ( ∩ (𝑉 × 𝑉))) = (𝑅 /s ( ∩ (𝑉 × 𝑉))))
13 eqid 2737 . . 3 (𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉))) = (𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉)))
14 inex1g 5212 . . . 4 ( 𝑊 → ( ∩ (𝑉 × 𝑉)) ∈ V)
159, 14syl 17 . . 3 (𝜑 → ( ∩ (𝑉 × 𝑉)) ∈ V)
1612, 7, 13, 15, 10qusval 17047 . 2 (𝜑 → (𝑅 /s ( ∩ (𝑉 × 𝑉))) = ((𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉))) “s 𝑅))
175, 11, 163eqtr4d 2787 1 (𝜑𝑈 = (𝑅 /s ( ∩ (𝑉 × 𝑉))))
Colors of variables: wff setvar class
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
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