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Theorem qusin 17525
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 8810 . . . . 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 7435 . 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 2728 . . 3 (𝑥𝑉 ↦ [𝑥] ) = (𝑥𝑉 ↦ [𝑥] )
9 qusin.e . . 3 (𝜑𝑊)
10 qusin.r . . 3 (𝜑𝑅𝑍)
116, 7, 8, 9, 10qusval 17523 . 2 (𝜑𝑈 = ((𝑥𝑉 ↦ [𝑥] ) “s 𝑅))
12 eqidd 2729 . . 3 (𝜑 → (𝑅 /s ( ∩ (𝑉 × 𝑉))) = (𝑅 /s ( ∩ (𝑉 × 𝑉))))
13 eqid 2728 . . 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 17523 . 2 (𝜑 → (𝑅 /s ( ∩ (𝑉 × 𝑉))) = ((𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉))) “s 𝑅))
175, 11, 163eqtr4d 2778 1 (𝜑𝑈 = (𝑅 /s ( ∩ (𝑉 × 𝑉))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  Vcvv 3471  cin 3946  wss 3947  cmpt 5231   × cxp 5676  cima 5681  cfv 6548  (class class class)co 7420  [cec 8722  Basecbs 17179  s cimas 17485   /s cqus 17486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-ec 8726  df-qus 17490
This theorem is referenced by:  pi1addf  24973  pi1addval  24974  pi1grplem  24975
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