Theorem dmuncnvepres 38890

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 5998	. 2 ⊢ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ dom (𝑅 ∪ ◡ E ))
2		dmun 5886	. . . 4 ⊢ dom (𝑅 ∪ ◡ E ) = (dom 𝑅 ∪ dom ◡ E )
3		dmcnvep 38887	. . . . 5 ⊢ dom ◡ E = (V ∖ {∅})
4	3	uneq2i 4118	. . . 4 ⊢ (dom 𝑅 ∪ dom ◡ E ) = (dom 𝑅 ∪ (V ∖ {∅}))
5	2, 4	eqtri 2785	. . 3 ⊢ dom (𝑅 ∪ ◡ E ) = (dom 𝑅 ∪ (V ∖ {∅}))
6	5	ineq2i 4169	. 2 ⊢ (𝐴 ∩ dom (𝑅 ∪ ◡ E )) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))
7	1, 6	eqtri 2785	1 ⊢ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))

Syntax hints:   = wceq 1560  Vcvv 3454   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903  ∅c0 4285  {csn 4582   E cep 5546  ^◡ccnv 5646  dom cdm 5647   ↾ cres 5649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-eprel 5547  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-res 5659

This theorem is referenced by:  dmxrnuncnvepres  38891  dfadjliftmap2  38956