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Theorem uniqs2 8773
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 8771 . . . . 5 (𝑅𝑉 (𝐴 / 𝑅) = (𝑅𝐴))
31, 2syl 17 . . . 4 (𝜑 (𝐴 / 𝑅) = (𝑅𝐴))
4 qsss.1 . . . . . 6 (𝜑𝑅 Er 𝐴)
5 erdm 8713 . . . . . 6 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
64, 5syl 17 . . . . 5 (𝜑 → dom 𝑅 = 𝐴)
76imaeq2d 6060 . . . 4 (𝜑 → (𝑅 “ dom 𝑅) = (𝑅𝐴))
83, 7eqtr4d 2776 . . 3 (𝜑 (𝐴 / 𝑅) = (𝑅 “ dom 𝑅))
9 imadmrn 6070 . . 3 (𝑅 “ dom 𝑅) = ran 𝑅
108, 9eqtrdi 2789 . 2 (𝜑 (𝐴 / 𝑅) = ran 𝑅)
11 errn 8725 . . 3 (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
124, 11syl 17 . 2 (𝜑 → ran 𝑅 = 𝐴)
1310, 12eqtrd 2773 1 (𝜑 (𝐴 / 𝑅) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107   cuni 4909  dom cdm 5677  ran crn 5678  cima 5680   Er wer 8700   / cqs 8702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-er 8703  df-ec 8705  df-qs 8709
This theorem is referenced by:  qshash  15773  cldsubg  23615  pi1buni  24556  qustrivr  32508
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