Theorem dmuncnvepres 38929

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 6012	. 2 ⊢ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ dom (𝑅 ∪ ◡ E ))
2		dmun 5901	. . . 4 ⊢ dom (𝑅 ∪ ◡ E ) = (dom 𝑅 ∪ dom ◡ E )
3		dmcnvep 38926	. . . . 5 ⊢ dom ◡ E = (V ∖ {∅})
4	3	uneq2i 4127	. . . 4 ⊢ (dom 𝑅 ∪ dom ◡ E ) = (dom 𝑅 ∪ (V ∖ {∅}))
5	2, 4	eqtri 2792	. . 3 ⊢ dom (𝑅 ∪ ◡ E ) = (dom 𝑅 ∪ (V ∖ {∅}))
6	5	ineq2i 4178	. 2 ⊢ (𝐴 ∩ dom (𝑅 ∪ ◡ E )) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))
7	1, 6	eqtri 2792	1 ⊢ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))

Syntax hints:   = wceq 1567  Vcvv 3463   ∖ cdif 3910   ∪ cun 3911   ∩ cin 3912  ∅c0 4294  {csn 4594   E cep 5561  ^◡ccnv 5661  dom cdm 5662   ↾ cres 5664

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-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

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-clab 2748  df-cleq 2761  df-clel 2844  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 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-eprel 5562  df-xp 5668  df-rel 5669  df-cnv 5670  df-dm 5672  df-res 5674

This theorem is referenced by:  dmxrnuncnvepres  38930  dfadjliftmap2  38995