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Theorem uniqs2 6102
Description: The union of a quotient set. (Contributed by Mario Carneiro, 11-Jul-2014.)
Hypotheses
Ref Expression
qsss.1 (φ𝑅 Er A)
qsss.2 (φ𝑅 𝑉)
Assertion
Ref Expression
uniqs2 (φ (A / 𝑅) = A)

Proof of Theorem uniqs2
StepHypRef Expression
1 qsss.2 . . . . 5 (φ𝑅 𝑉)
2 uniqs 6100 . . . . 5 (𝑅 𝑉 (A / 𝑅) = (𝑅A))
31, 2syl 14 . . . 4 (φ (A / 𝑅) = (𝑅A))
4 qsss.1 . . . . . 6 (φ𝑅 Er A)
5 erdm 6052 . . . . . 6 (𝑅 Er A → dom 𝑅 = A)
64, 5syl 14 . . . . 5 (φ → dom 𝑅 = A)
76imaeq2d 4611 . . . 4 (φ → (𝑅 “ dom 𝑅) = (𝑅A))
83, 7eqtr4d 2072 . . 3 (φ (A / 𝑅) = (𝑅 “ dom 𝑅))
9 imadmrn 4621 . . 3 (𝑅 “ dom 𝑅) = ran 𝑅
108, 9syl6eq 2085 . 2 (φ (A / 𝑅) = ran 𝑅)
11 errn 6064 . . 3 (𝑅 Er A → ran 𝑅 = A)
124, 11syl 14 . 2 (φ → ran 𝑅 = A)
1310, 12eqtrd 2069 1 (φ (A / 𝑅) = A)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390   cuni 3571  dom cdm 4288  ran crn 4289  cima 4291   Er wer 6039   / cqs 6041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-er 6042  df-ec 6044  df-qs 6048
This theorem is referenced by: (None)
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