Theorem eldmressnALTV 39131

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 39129	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝐵 ∈ {𝐴} ∧ ∃𝑦 𝐵𝑅𝑦)))
2		elsng 4597	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ {𝐴} ↔ 𝐵 = 𝐴))
3		eldmg 5876	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom 𝑅 ↔ ∃𝑦 𝐵𝑅𝑦))
4	3	bicomd 226	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (∃𝑦 𝐵𝑅𝑦 ↔ 𝐵 ∈ dom 𝑅))
5	2, 4	anbi12d 644	. . 3 ⊢ (𝐵 ∈ 𝑉 → ((𝐵 ∈ {𝐴} ∧ ∃𝑦 𝐵𝑅𝑦) ↔ (𝐵 = 𝐴 ∧ 𝐵 ∈ dom 𝑅)))
6	1, 5	bitrd 282	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝐵 = 𝐴 ∧ 𝐵 ∈ dom 𝑅)))
7		eqelb 39093	. 2 ⊢ ((𝐵 = 𝐴 ∧ 𝐵 ∈ dom 𝑅) ↔ (𝐵 = 𝐴 ∧ 𝐴 ∈ dom 𝑅))
8	6, 7	bitrdi 290	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝐵 = 𝐴 ∧ 𝐴 ∈ dom 𝑅)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {csn 4583   class class class wbr 5102  dom cdm 5647   ↾ cres 5649

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5653  df-dm 5657  df-res 5659

This theorem is used by:  refressn  39385