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Theorem uniqs 7962
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 7903 . . . . 5 (𝑅𝑉 → [𝑥]𝑅 ∈ V)
21ralrimivw 3093 . . . 4 (𝑅𝑉 → ∀𝑥𝐴 [𝑥]𝑅 ∈ V)
3 dfiun2g 4692 . . . 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 2754 . 2 (𝑅𝑉 {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅} = 𝑥𝐴 [𝑥]𝑅)
6 df-qs 7905 . . 3 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
76unieqi 4585 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
8 df-ec 7901 . . . . 5 [𝑥]𝑅 = (𝑅 “ {𝑥})
98a1i 11 . . . 4 (𝑥𝐴 → [𝑥]𝑅 = (𝑅 “ {𝑥}))
109iuneq2i 4679 . . 3 𝑥𝐴 [𝑥]𝑅 = 𝑥𝐴 (𝑅 “ {𝑥})
11 imaiun 6654 . . 3 (𝑅 𝑥𝐴 {𝑥}) = 𝑥𝐴 (𝑅 “ {𝑥})
12 iunid 4715 . . . 4 𝑥𝐴 {𝑥} = 𝐴
1312imaeq2i 5610 . . 3 (𝑅 𝑥𝐴 {𝑥}) = (𝑅𝐴)
1410, 11, 133eqtr2ri 2777 . 2 (𝑅𝐴) = 𝑥𝐴 [𝑥]𝑅
155, 7, 143eqtr4g 2807 1 (𝑅𝑉 (𝐴 / 𝑅) = (𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1620  wcel 2127  {cab 2734  wral 3038  wrex 3039  Vcvv 3328  {csn 4309   cuni 4576   ciun 4660  cima 5257  [cec 7897   / cqs 7898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-xp 5260  df-cnv 5262  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-ec 7901  df-qs 7905
This theorem is referenced by:  uniqs2  7964  ecqs  7966
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