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Theorem dm1cosscnvepres 38437
Description: The domain of cosets of the restricted converse epsilon relation is the union of the restriction. (Contributed by Peter Mazsa, 18-May-2019.) (Revised by Peter Mazsa, 26-Sep-2021.)
Assertion
Ref Expression
dm1cosscnvepres dom ≀ ( E ↾ 𝐴) = 𝐴

Proof of Theorem dm1cosscnvepres
StepHypRef Expression
1 dmcoss2 38435 . 2 dom ≀ ( E ↾ 𝐴) = ran ( E ↾ 𝐴)
2 rncnvepres 38284 . 2 ran ( E ↾ 𝐴) = 𝐴
31, 2eqtri 2762 1 dom ≀ ( E ↾ 𝐴) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536   cuni 4911   E cep 5587  ccnv 5687  dom cdm 5688  ran crn 5689  cres 5690  ccoss 38161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-eprel 5588  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-coss 38392
This theorem is referenced by:  dmcoels  38438
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