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Theorem uniqs 6453
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 6399 . . . . 5 (𝑅𝑉 → [𝑥]𝑅 ∈ V)
21ralrimivw 2481 . . . 4 (𝑅𝑉 → ∀𝑥𝐴 [𝑥]𝑅 ∈ V)
3 dfiun2g 3813 . . . 4 (∀𝑥𝐴 [𝑥]𝑅 ∈ V → 𝑥𝐴 [𝑥]𝑅 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅})
42, 3syl 14 . . 3 (𝑅𝑉 𝑥𝐴 [𝑥]𝑅 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅})
54eqcomd 2121 . 2 (𝑅𝑉 {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅} = 𝑥𝐴 [𝑥]𝑅)
6 df-qs 6401 . . 3 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
76unieqi 3714 . 2 (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝑅}
8 df-ec 6397 . . . . 5 [𝑥]𝑅 = (𝑅 “ {𝑥})
98a1i 9 . . . 4 (𝑥𝐴 → [𝑥]𝑅 = (𝑅 “ {𝑥}))
109iuneq2i 3799 . . 3 𝑥𝐴 [𝑥]𝑅 = 𝑥𝐴 (𝑅 “ {𝑥})
11 imaiun 5627 . . 3 (𝑅 𝑥𝐴 {𝑥}) = 𝑥𝐴 (𝑅 “ {𝑥})
12 iunid 3836 . . . 4 𝑥𝐴 {𝑥} = 𝐴
1312imaeq2i 4847 . . 3 (𝑅 𝑥𝐴 {𝑥}) = (𝑅𝐴)
1410, 11, 133eqtr2ri 2143 . 2 (𝑅𝐴) = 𝑥𝐴 [𝑥]𝑅
155, 7, 143eqtr4g 2173 1 (𝑅𝑉 (𝐴 / 𝑅) = (𝑅𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1314  wcel 1463  {cab 2101  wral 2391  wrex 2392  Vcvv 2658  {csn 3495   cuni 3704   ciun 3781  cima 4510  [cec 6393   / cqs 6394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-xp 4513  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-ec 6397  df-qs 6401
This theorem is referenced by:  uniqs2  6455  ecqs  6457
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