Theorem eldmressnALTV 36482

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 36480	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝐵 ∈ {𝐴} ∧ ∃𝑦 𝐵𝑅𝑦)))
2		elsng 4579	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ {𝐴} ↔ 𝐵 = 𝐴))
3		eldmg 5820	. . . . 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 36440	. 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 1539  ∃wex 1779   ∈ wcel 2104  {csn 4565   class class class wbr 5081  dom cdm 5600   ↾ cres 5602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3333  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-xp 5606  df-dm 5610  df-res 5612

This theorem is referenced by:  refressn  36657