Theorem dm1cosscnvepres 39080

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 39078	. 2 ⊢ dom ≀ (◡ E ↾ 𝐴) = ran (◡ E ↾ 𝐴)
2		rncnvepres 38843	. 2 ⊢ ran (◡ E ↾ 𝐴) = ∪ 𝐴
3	1, 2	eqtri 2792	1 ⊢ dom ≀ (◡ E ↾ 𝐴) = ∪ 𝐴

Syntax hints:   = wceq 1567  ∪ cuni 4873   E cep 5558  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   ↾ cres 5661   ≀ ccoss 38717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-eprel 5559  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-coss 39035

This theorem is referenced by:  dmcoels  39081