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Theorem uniqs2 6457
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 6455 . . . . 5 (𝑅𝑉 (𝐴 / 𝑅) = (𝑅𝐴))
31, 2syl 14 . . . 4 (𝜑 (𝐴 / 𝑅) = (𝑅𝐴))
4 qsss.1 . . . . . 6 (𝜑𝑅 Er 𝐴)
5 erdm 6407 . . . . . 6 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
64, 5syl 14 . . . . 5 (𝜑 → dom 𝑅 = 𝐴)
76imaeq2d 4851 . . . 4 (𝜑 → (𝑅 “ dom 𝑅) = (𝑅𝐴))
83, 7eqtr4d 2153 . . 3 (𝜑 (𝐴 / 𝑅) = (𝑅 “ dom 𝑅))
9 imadmrn 4861 . . 3 (𝑅 “ dom 𝑅) = ran 𝑅
108, 9syl6eq 2166 . 2 (𝜑 (𝐴 / 𝑅) = ran 𝑅)
11 errn 6419 . . 3 (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
124, 11syl 14 . 2 (𝜑 → ran 𝑅 = 𝐴)
1310, 12eqtrd 2150 1 (𝜑 (𝐴 / 𝑅) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wcel 1465   cuni 3706  dom cdm 4509  ran crn 4510  cima 4512   Er wer 6394   / cqs 6396
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-er 6397  df-ec 6399  df-qs 6403
This theorem is referenced by: (None)
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