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Theorem uniqs 8796
Description: The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
Assertion
Ref Expression
uniqs (𝑅𝑉 (𝐴 / 𝑅) = (𝑅𝐴))

Proof of Theorem uniqs
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecexg 8729 . . . . 5 (𝑅𝑉 → [𝑥]𝑅 ∈ V)
21ralrimivw 3139 . . . 4 (𝑅𝑉 → ∀𝑥𝐴 [𝑥]𝑅 ∈ V)
3 dfiun2g 5034 . . . 4 (∀𝑥𝐴 [𝑥]𝑅 ∈ V → 𝑥𝐴 [𝑥]𝑅 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅})
42, 3syl 17 . . 3 (𝑅𝑉 𝑥𝐴 [𝑥]𝑅 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅})
54eqcomd 2731 . 2 (𝑅𝑉 {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅} = 𝑥𝐴 [𝑥]𝑅)
6 df-qs 8731 . . 3 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
76unieqi 4921 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
8 df-ec 8727 . . . . 5 [𝑥]𝑅 = (𝑅 “ {𝑥})
98a1i 11 . . . 4 (𝑥𝐴 → [𝑥]𝑅 = (𝑅 “ {𝑥}))
109iuneq2i 5018 . . 3 𝑥𝐴 [𝑥]𝑅 = 𝑥𝐴 (𝑅 “ {𝑥})
11 imaiun 7255 . . 3 (𝑅 𝑥𝐴 {𝑥}) = 𝑥𝐴 (𝑅 “ {𝑥})
12 iunid 5064 . . . 4 𝑥𝐴 {𝑥} = 𝐴
1312imaeq2i 6062 . . 3 (𝑅 𝑥𝐴 {𝑥}) = (𝑅𝐴)
1410, 11, 133eqtr2ri 2760 . 2 (𝑅𝐴) = 𝑥𝐴 [𝑥]𝑅
155, 7, 143eqtr4g 2790 1 (𝑅𝑉 (𝐴 / 𝑅) = (𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {cab 2702  wral 3050  wrex 3059  Vcvv 3461  {csn 4630   cuni 4909   ciun 4997  cima 5681  [cec 8723   / cqs 8724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-xp 5684  df-cnv 5686  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8727  df-qs 8731
This theorem is referenced by:  uniqs2  8798  ecqs  8800
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