Theorem dmuncnvepres 39589

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 6020	. 2 ⊢ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ dom (𝑅 ∪ ◡ E ))
2		dmun 5904	. . . 4 ⊢ dom (𝑅 ∪ ◡ E ) = (dom 𝑅 ∪ dom ◡ E )
3		dmcnvep 39586	. . . . 5 ⊢ dom ◡ E = (V ∖ {∅})
4	3	uneq2i 4167	. . . 4 ⊢ (dom 𝑅 ∪ dom ◡ E ) = (dom 𝑅 ∪ (V ∖ {∅}))
5	2, 4	eqtri 2786	. . 3 ⊢ dom (𝑅 ∪ ◡ E ) = (dom 𝑅 ∪ (V ∖ {∅}))
6	5	ineq2i 4117	. 2 ⊢ (𝐴 ∩ dom (𝑅 ∪ ◡ E )) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))
7	1, 6	eqtri 2786	1 ⊢ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3454   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588   E cep 5557  ^◡ccnv 5657  dom cdm 5658   ↾ cres 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-eprel 5558  df-xp 5664  df-rel 5665  df-cnv 5666  df-dm 5668  df-res 5670

This theorem is used by:  dmxrnuncnvepres  39590  dfadjliftmap2  39655