Theorem dm1cosscnvepres 38564

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 38562	. 2 ⊢ dom ≀ (◡ E ↾ 𝐴) = ran (◡ E ↾ 𝐴)
2		rncnvepres 38347	. 2 ⊢ ran (◡ E ↾ 𝐴) = ∪ 𝐴
3	1, 2	eqtri 2754	1 ⊢ dom ≀ (◡ E ↾ 𝐴) = ∪ 𝐴

Syntax hints:   = wceq 1541  ∪ cuni 4858   E cep 5518  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   ↾ cres 5621   ≀ ccoss 38228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-eprel 5519  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-coss 38519

This theorem is referenced by:  dmcoels  38565