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Theorem uniqs 6063
Description: The union of a quotient set. (Contributed by NM, 9-Dec-2008.)
Assertion
Ref Expression
uniqs (𝑅 𝑉 (A / 𝑅) = (𝑅A))

Proof of Theorem uniqs
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecexg 6009 . . . . 5 (𝑅 𝑉 → [x]𝑅 V)
21ralrimivw 2362 . . . 4 (𝑅 𝑉x A [x]𝑅 V)
3 dfiun2g 3652 . . . 4 (x A [x]𝑅 V → x A [x]𝑅 = {yx A y = [x]𝑅})
42, 3syl 14 . . 3 (𝑅 𝑉 x A [x]𝑅 = {yx A y = [x]𝑅})
54eqcomd 2018 . 2 (𝑅 𝑉 {yx A y = [x]𝑅} = x A [x]𝑅)
6 df-qs 6011 . . 3 (A / 𝑅) = {yx A y = [x]𝑅}
76unieqi 3553 . 2 (A / 𝑅) = {yx A y = [x]𝑅}
8 df-ec 6007 . . . . 5 [x]𝑅 = (𝑅 “ {x})
98a1i 9 . . . 4 (x A → [x]𝑅 = (𝑅 “ {x}))
109iuneq2i 3638 . . 3 x A [x]𝑅 = x A (𝑅 “ {x})
11 imaiun 5312 . . 3 (𝑅 x A {x}) = x A (𝑅 “ {x})
12 iunid 3675 . . . 4 x A {x} = A
1312imaeq2i 4581 . . 3 (𝑅 x A {x}) = (𝑅A)
1410, 11, 133eqtr2ri 2040 . 2 (𝑅A) = x A [x]𝑅
155, 7, 143eqtr4g 2070 1 (𝑅 𝑉 (A / 𝑅) = (𝑅A))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1223   wcel 1366  {cab 1999  wral 2275  wrex 2276  Vcvv 2526  {csn 3339   cuni 3543   ciun 3620  cima 4263  [cec 6003   / cqs 6004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-13 1377  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907  ax-un 4108
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-iun 3622  df-br 3728  df-opab 3782  df-xp 4266  df-cnv 4268  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-ec 6007  df-qs 6011
This theorem is referenced by:  uniqs2  6065  ecqs  6067
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