Theorem dmcnvepres 39039

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 6011	. 2 ⊢ dom (◡ E ↾ 𝐴) = (𝐴 ∩ dom ◡ E )
2		dmcnvep 39037	. . 3 ⊢ dom ◡ E = (V ∖ {∅})
3	2	ineq2i 4170	. 2 ⊢ (𝐴 ∩ dom ◡ E ) = (𝐴 ∩ (V ∖ {∅}))
4		invdif 4232	. 2 ⊢ (𝐴 ∩ (V ∖ {∅})) = (𝐴 ∖ {∅})
5	1, 3, 4	3eqtri 2790	1 ⊢ dom (◡ E ↾ 𝐴) = (𝐴 ∖ {∅})

Syntax hints:   = wceq 1570  Vcvv 3455   ∖ cdif 3902   ∩ 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: (None)