Theorem dm1cosscnvepres 36501

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

Step	Hyp	Ref	Expression
1		dmcoss2 36499	. 2 ⊢ dom ≀ (◡ E ↾ 𝐴) = ran (◡ E ↾ 𝐴)
2		rncnvepres 36366	. 2 ⊢ ran (◡ E ↾ 𝐴) = ∪ 𝐴
3	1, 2	eqtri 2766	1 ⊢ dom ≀ (◡ E ↾ 𝐴) = ∪ 𝐴

Syntax hints:   = wceq 1539  ∪ cuni 4836   E cep 5485  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   ↾ cres 5582   ≀ ccoss 36260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-eprel 5486  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-coss 36464

This theorem is referenced by:  dmcoels  36502