Theorem dm1cosscnvepres 38405

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 38403	. 2 ⊢ dom ≀ (◡ E ↾ 𝐴) = ran (◡ E ↾ 𝐴)
2		rncnvepres 38252	. 2 ⊢ ran (◡ E ↾ 𝐴) = ∪ 𝐴
3	1, 2	eqtri 2768	1 ⊢ dom ≀ (◡ E ↾ 𝐴) = ∪ 𝐴

Syntax hints:   = wceq 1537  ∪ cuni 4931   E cep 5598  ^◡ccnv 5694  dom cdm 5695  ran crn 5696   ↾ cres 5697   ≀ ccoss 38128

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-eprel 5599  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-coss 38360

This theorem is referenced by:  dmcoels  38406