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Theorem uniqs2 7960
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 7958 . . . . 5 (𝑅𝑉 (𝐴 / 𝑅) = (𝑅𝐴))
31, 2syl 17 . . . 4 (𝜑 (𝐴 / 𝑅) = (𝑅𝐴))
4 qsss.1 . . . . . 6 (𝜑𝑅 Er 𝐴)
5 erdm 7905 . . . . . 6 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
64, 5syl 17 . . . . 5 (𝜑 → dom 𝑅 = 𝐴)
76imaeq2d 5607 . . . 4 (𝜑 → (𝑅 “ dom 𝑅) = (𝑅𝐴))
83, 7eqtr4d 2807 . . 3 (𝜑 (𝐴 / 𝑅) = (𝑅 “ dom 𝑅))
9 imadmrn 5617 . . 3 (𝑅 “ dom 𝑅) = ran 𝑅
108, 9syl6eq 2820 . 2 (𝜑 (𝐴 / 𝑅) = ran 𝑅)
11 errn 7917 . . 3 (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
124, 11syl 17 . 2 (𝜑 → ran 𝑅 = 𝐴)
1310, 12eqtrd 2804 1 (𝜑 (𝐴 / 𝑅) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144   cuni 4572  dom cdm 5249  ran crn 5250  cima 5252   Er wer 7892   / cqs 7894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-er 7895  df-ec 7897  df-qs 7901
This theorem is referenced by:  qshash  14765  cldsubg  22133  pi1buni  23058
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