Theorem dmuncnvepres 39040

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 6011	. 2 ⊢ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ dom (𝑅 ∪ ◡ E ))
2		dmun 5900	. . . 4 ⊢ dom (𝑅 ∪ ◡ E ) = (dom 𝑅 ∪ dom ◡ E )
3		dmcnvep 39037	. . . . 5 ⊢ dom ◡ E = (V ∖ {∅})
4	3	uneq2i 4119	. . . 4 ⊢ (dom 𝑅 ∪ dom ◡ E ) = (dom 𝑅 ∪ (V ∖ {∅}))
5	2, 4	eqtri 2786	. . 3 ⊢ dom (𝑅 ∪ ◡ E ) = (dom 𝑅 ∪ (V ∖ {∅}))
6	5	ineq2i 4170	. 2 ⊢ (𝐴 ∩ dom (𝑅 ∪ ◡ E )) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))
7	1, 6	eqtri 2786	1 ⊢ dom ((𝑅 ∪ ◡ E ) ↾ 𝐴) = (𝐴 ∩ (dom 𝑅 ∪ (V ∖ {∅})))

Syntax hints:   = wceq 1570  Vcvv 3455   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904  ∅c0 4286  {csn 4589   E cep 5560  ^◡ccnv 5660  dom cdm 5661   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-res 5673

This theorem is referenced by:  dmxrnuncnvepres  39041  dfadjliftmap2  39106