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Theorem qusin 16185
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 7807 . . . . 5 ((( 𝑉) ⊆ 𝑉𝑥𝑉) → [𝑥] = [𝑥]( ∩ (𝑉 × 𝑉)))
31, 2sylan 488 . . . 4 ((𝜑𝑥𝑉) → [𝑥] = [𝑥]( ∩ (𝑉 × 𝑉)))
43mpteq2dva 4735 . . 3 (𝜑 → (𝑥𝑉 ↦ [𝑥] ) = (𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉))))
54oveq1d 6650 . 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 2620 . . 3 (𝑥𝑉 ↦ [𝑥] ) = (𝑥𝑉 ↦ [𝑥] )
9 qusin.e . . 3 (𝜑𝑊)
10 qusin.r . . 3 (𝜑𝑅𝑍)
116, 7, 8, 9, 10qusval 16183 . 2 (𝜑𝑈 = ((𝑥𝑉 ↦ [𝑥] ) “s 𝑅))
12 eqidd 2621 . . 3 (𝜑 → (𝑅 /s ( ∩ (𝑉 × 𝑉))) = (𝑅 /s ( ∩ (𝑉 × 𝑉))))
13 eqid 2620 . . 3 (𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉))) = (𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉)))
14 inex1g 4792 . . . 4 ( 𝑊 → ( ∩ (𝑉 × 𝑉)) ∈ V)
159, 14syl 17 . . 3 (𝜑 → ( ∩ (𝑉 × 𝑉)) ∈ V)
1612, 7, 13, 15, 10qusval 16183 . 2 (𝜑 → (𝑅 /s ( ∩ (𝑉 × 𝑉))) = ((𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉))) “s 𝑅))
175, 11, 163eqtr4d 2664 1 (𝜑𝑈 = (𝑅 /s ( ∩ (𝑉 × 𝑉))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  Vcvv 3195  cin 3566  wss 3567  cmpt 4720   × cxp 5102  cima 5107  cfv 5876  (class class class)co 6635  [cec 7725  Basecbs 15838  s cimas 16145   /s cqus 16146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-ec 7729  df-qus 16150
This theorem is referenced by:  pi1addf  22828  pi1addval  22829  pi1grplem  22830
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