Theorem dmcnvepres 38711

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

Assertion

Ref	Expression
dmcnvepres	⊢ dom (◡ E ↾ 𝐴) = (𝐴 ∖ {∅})

Proof of Theorem dmcnvepres

Step	Hyp	Ref	Expression
1		dmres 5977	. 2 ⊢ dom (◡ E ↾ 𝐴) = (𝐴 ∩ dom ◡ E )
2		dmcnvep 38709	. . 3 ⊢ dom ◡ E = (V ∖ {∅})
3	2	ineq2i 4157	. 2 ⊢ (𝐴 ∩ dom ◡ E ) = (𝐴 ∩ (V ∖ {∅}))
4		invdif 4219	. 2 ⊢ (𝐴 ∩ (V ∖ {∅})) = (𝐴 ∖ {∅})
5	1, 3, 4	3eqtri 2763	1 ⊢ dom (◡ E ↾ 𝐴) = (𝐴 ∖ {∅})

Syntax hints:   = wceq 1542  Vcvv 3429   ∖ cdif 3886   ∩ cin 3888  ∅c0 4273  {csn 4567   E cep 5530  ^◡ccnv 5630  dom cdm 5631   ↾ cres 5633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-eprel 5531  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-res 5643

This theorem is referenced by: (None)