MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniqs2 Structured version   Visualization version   GIF version

Theorem uniqs2 8819
Description: The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)
Hypotheses
Ref Expression
qsss.1 (𝜑𝑅 Er 𝐴)
qsss.2 (𝜑𝑅𝑉)
Assertion
Ref Expression
uniqs2 (𝜑 (𝐴 / 𝑅) = 𝐴)

Proof of Theorem uniqs2
StepHypRef Expression
1 qsss.2 . . . . 5 (𝜑𝑅𝑉)
2 uniqs 8817 . . . . 5 (𝑅𝑉 (𝐴 / 𝑅) = (𝑅𝐴))
31, 2syl 17 . . . 4 (𝜑 (𝐴 / 𝑅) = (𝑅𝐴))
4 qsss.1 . . . . . 6 (𝜑𝑅 Er 𝐴)
5 erdm 8755 . . . . . 6 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
64, 5syl 17 . . . . 5 (𝜑 → dom 𝑅 = 𝐴)
76imaeq2d 6078 . . . 4 (𝜑 → (𝑅 “ dom 𝑅) = (𝑅𝐴))
83, 7eqtr4d 2780 . . 3 (𝜑 (𝐴 / 𝑅) = (𝑅 “ dom 𝑅))
9 imadmrn 6088 . . 3 (𝑅 “ dom 𝑅) = ran 𝑅
108, 9eqtrdi 2793 . 2 (𝜑 (𝐴 / 𝑅) = ran 𝑅)
11 errn 8767 . . 3 (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
124, 11syl 17 . 2 (𝜑 → ran 𝑅 = 𝐴)
1310, 12eqtrd 2777 1 (𝜑 (𝐴 / 𝑅) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108   cuni 4907  dom cdm 5685  ran crn 5686  cima 5688   Er wer 8742   / cqs 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-er 8745  df-ec 8747  df-qs 8751
This theorem is referenced by:  qshash  15863  cldsubg  24119  pi1buni  25073  qustrivr  33393
  Copyright terms: Public domain W3C validator