Theorem eldmressnALTV 37129

Description: Element of the domain of a restriction to a singleton. (Contributed by Peter Mazsa, 12-Jun-2024.)

Assertion

Ref	Expression
eldmressnALTV	⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝐵 = 𝐴 ∧ 𝐴 ∈ dom 𝑅)))

Proof of Theorem eldmressnALTV

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldmres 37127	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝐵 ∈ {𝐴} ∧ ∃𝑦 𝐵𝑅𝑦)))
2		elsng 4642	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ {𝐴} ↔ 𝐵 = 𝐴))
3		eldmg 5897	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom 𝑅 ↔ ∃𝑦 𝐵𝑅𝑦))
4	3	bicomd 222	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (∃𝑦 𝐵𝑅𝑦 ↔ 𝐵 ∈ dom 𝑅))
5	2, 4	anbi12d 632	. . 3 ⊢ (𝐵 ∈ 𝑉 → ((𝐵 ∈ {𝐴} ∧ ∃𝑦 𝐵𝑅𝑦) ↔ (𝐵 = 𝐴 ∧ 𝐵 ∈ dom 𝑅)))
6	1, 5	bitrd 279	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝐵 = 𝐴 ∧ 𝐵 ∈ dom 𝑅)))
7		eqelb 37090	. 2 ⊢ ((𝐵 = 𝐴 ∧ 𝐵 ∈ dom 𝑅) ↔ (𝐵 = 𝐴 ∧ 𝐴 ∈ dom 𝑅))
8	6, 7	bitrdi 287	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝐵 = 𝐴 ∧ 𝐴 ∈ dom 𝑅)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  {csn 4628   class class class wbr 5148  dom cdm 5676   ↾ cres 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-dm 5686  df-res 5688

This theorem is referenced by:  refressn  37302