Theorem eldmressnALTV 38254

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 38252	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝐵 ∈ {𝐴} ∧ ∃𝑦 𝐵𝑅𝑦)))
2		elsng 4645	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ {𝐴} ↔ 𝐵 = 𝐴))
3		eldmg 5912	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom 𝑅 ↔ ∃𝑦 𝐵𝑅𝑦))
4	3	bicomd 223	. . . 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 38216	. 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 206   ∧ wa 395   = wceq 1537  ∃wex 1776   ∈ wcel 2106  {csn 4631   class class class wbr 5148  dom cdm 5689   ↾ cres 5691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-dm 5699  df-res 5701

This theorem is referenced by:  refressn  38425