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Theorem qusin 17497
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 8792 . . . . 5 ((( 𝑉) ⊆ 𝑉𝑥𝑉) → [𝑥] = [𝑥]( ∩ (𝑉 × 𝑉)))
31, 2sylan 579 . . . 4 ((𝜑𝑥𝑉) → [𝑥] = [𝑥]( ∩ (𝑉 × 𝑉)))
43mpteq2dva 5248 . . 3 (𝜑 → (𝑥𝑉 ↦ [𝑥] ) = (𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉))))
54oveq1d 7427 . 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 2731 . . 3 (𝑥𝑉 ↦ [𝑥] ) = (𝑥𝑉 ↦ [𝑥] )
9 qusin.e . . 3 (𝜑𝑊)
10 qusin.r . . 3 (𝜑𝑅𝑍)
116, 7, 8, 9, 10qusval 17495 . 2 (𝜑𝑈 = ((𝑥𝑉 ↦ [𝑥] ) “s 𝑅))
12 eqidd 2732 . . 3 (𝜑 → (𝑅 /s ( ∩ (𝑉 × 𝑉))) = (𝑅 /s ( ∩ (𝑉 × 𝑉))))
13 eqid 2731 . . 3 (𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉))) = (𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉)))
14 inex1g 5319 . . . 4 ( 𝑊 → ( ∩ (𝑉 × 𝑉)) ∈ V)
159, 14syl 17 . . 3 (𝜑 → ( ∩ (𝑉 × 𝑉)) ∈ V)
1612, 7, 13, 15, 10qusval 17495 . 2 (𝜑 → (𝑅 /s ( ∩ (𝑉 × 𝑉))) = ((𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉))) “s 𝑅))
175, 11, 163eqtr4d 2781 1 (𝜑𝑈 = (𝑅 /s ( ∩ (𝑉 × 𝑉))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2105  Vcvv 3473  cin 3947  wss 3948  cmpt 5231   × cxp 5674  cima 5679  cfv 6543  (class class class)co 7412  [cec 8707  Basecbs 17151  s cimas 17457   /s cqus 17458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-ec 8711  df-qus 17462
This theorem is referenced by:  pi1addf  24894  pi1addval  24895  pi1grplem  24896
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