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Theorem qusin 16829
 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 8373 . . . . 5 ((( 𝑉) ⊆ 𝑉𝑥𝑉) → [𝑥] = [𝑥]( ∩ (𝑉 × 𝑉)))
31, 2sylan 583 . . . 4 ((𝜑𝑥𝑉) → [𝑥] = [𝑥]( ∩ (𝑉 × 𝑉)))
43mpteq2dva 5129 . . 3 (𝜑 → (𝑥𝑉 ↦ [𝑥] ) = (𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉))))
54oveq1d 7160 . 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 2798 . . 3 (𝑥𝑉 ↦ [𝑥] ) = (𝑥𝑉 ↦ [𝑥] )
9 qusin.e . . 3 (𝜑𝑊)
10 qusin.r . . 3 (𝜑𝑅𝑍)
116, 7, 8, 9, 10qusval 16827 . 2 (𝜑𝑈 = ((𝑥𝑉 ↦ [𝑥] ) “s 𝑅))
12 eqidd 2799 . . 3 (𝜑 → (𝑅 /s ( ∩ (𝑉 × 𝑉))) = (𝑅 /s ( ∩ (𝑉 × 𝑉))))
13 eqid 2798 . . 3 (𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉))) = (𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉)))
14 inex1g 5191 . . . 4 ( 𝑊 → ( ∩ (𝑉 × 𝑉)) ∈ V)
159, 14syl 17 . . 3 (𝜑 → ( ∩ (𝑉 × 𝑉)) ∈ V)
1612, 7, 13, 15, 10qusval 16827 . 2 (𝜑 → (𝑅 /s ( ∩ (𝑉 × 𝑉))) = ((𝑥𝑉 ↦ [𝑥]( ∩ (𝑉 × 𝑉))) “s 𝑅))
175, 11, 163eqtr4d 2843 1 (𝜑𝑈 = (𝑅 /s ( ∩ (𝑉 × 𝑉))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3442   ∩ cin 3882   ⊆ wss 3883   ↦ cmpt 5114   × cxp 5521   “ cima 5526  ‘cfv 6332  (class class class)co 7145  [cec 8288  Basecbs 16495   “s cimas 16789   /s cqus 16790 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-ec 8292  df-qus 16794 This theorem is referenced by:  pi1addf  23693  pi1addval  23694  pi1grplem  23695
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