Theorem dm1cosscnvepres 38457

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 38455	. 2 ⊢ dom ≀ (◡ E ↾ 𝐴) = ran (◡ E ↾ 𝐴)
2		rncnvepres 38304	. 2 ⊢ ran (◡ E ↾ 𝐴) = ∪ 𝐴
3	1, 2	eqtri 2765	1 ⊢ dom ≀ (◡ E ↾ 𝐴) = ∪ 𝐴

Syntax hints:   = wceq 1540  ∪ cuni 4907   E cep 5583  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   ↾ cres 5687   ≀ ccoss 38182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-coss 38412

This theorem is referenced by:  dmcoels  38458