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Theorem uniqs2 6254
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 6252 . . . . 5 (𝑅𝑉 (𝐴 / 𝑅) = (𝑅𝐴))
31, 2syl 14 . . . 4 (𝜑 (𝐴 / 𝑅) = (𝑅𝐴))
4 qsss.1 . . . . . 6 (𝜑𝑅 Er 𝐴)
5 erdm 6204 . . . . . 6 (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
64, 5syl 14 . . . . 5 (𝜑 → dom 𝑅 = 𝐴)
76imaeq2d 4718 . . . 4 (𝜑 → (𝑅 “ dom 𝑅) = (𝑅𝐴))
83, 7eqtr4d 2118 . . 3 (𝜑 (𝐴 / 𝑅) = (𝑅 “ dom 𝑅))
9 imadmrn 4728 . . 3 (𝑅 “ dom 𝑅) = ran 𝑅
108, 9syl6eq 2131 . 2 (𝜑 (𝐴 / 𝑅) = ran 𝑅)
11 errn 6216 . . 3 (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)
124, 11syl 14 . 2 (𝜑 → ran 𝑅 = 𝐴)
1310, 12eqtrd 2115 1 (𝜑 (𝐴 / 𝑅) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434   cuni 3621  dom cdm 4391  ran crn 4392  cima 4394   Er wer 6191   / cqs 6193
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-er 6194  df-ec 6196  df-qs 6200
This theorem is referenced by: (None)
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