Theorem dmuncnvepres 38515

Description: Domain of the union with the converse epsilon, restricted. (Contributed by Peter Mazsa, 28-Jan-2026.)

Assertion

Ref	Expression
dmuncnvepres	⊢ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))

Proof of Theorem dmuncnvepres

Step	Hyp	Ref	Expression
1		dmres 5969	. 2 ⊢ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ dom (𝑅 ∪ ◡ E ))
2		dmun 5857	. . . 4 ⊢ dom (𝑅 ∪ ◡ E ) = (dom 𝑅 ∪ dom ◡ E )
3		dmcnvep 38512	. . . . 5 ⊢ dom ◡ E = (V ∖ {∅})
4	3	uneq2i 4115	. . . 4 ⊢ (dom 𝑅 ∪ dom ◡ E ) = (dom 𝑅 ∪ (V ∖ {∅}))
5	2, 4	eqtri 2757	. . 3 ⊢ dom (𝑅 ∪ ◡ E ) = (dom 𝑅 ∪ (V ∖ {∅}))
6	5	ineq2i 4167	. 2 ⊢ (𝐴 ∩ dom (𝑅 ∪ ◡ E )) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))
7	1, 6	eqtri 2757	1 ⊢ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))

Syntax hints:   = wceq 1541  Vcvv 3438   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898  ∅c0 4283  {csn 4578   E cep 5521  ^◡ccnv 5621  dom cdm 5622   ↾ cres 5624

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 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

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-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-eprel 5522  df-xp 5628  df-rel 5629  df-cnv 5630  df-dm 5632  df-res 5634

This theorem is referenced by:  dmxrnuncnvepres  38516  dfadjliftmap2  38571