Theorem eldmressnALTV 38726

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 38724	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝐵 ∈ {𝐴} ∧ ∃𝑦 𝐵𝑅𝑦)))
2		elsng 4590	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ {𝐴} ↔ 𝐵 = 𝐴))
3		eldmg 5867	. . . . 5 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom 𝑅 ↔ ∃𝑦 𝐵𝑅𝑦))
4	3	bicomd 225	. . . 4 ⊢ (𝐵 ∈ 𝑉 → (∃𝑦 𝐵𝑅𝑦 ↔ 𝐵 ∈ dom 𝑅))
5	2, 4	anbi12d 640	. . 3 ⊢ (𝐵 ∈ 𝑉 → ((𝐵 ∈ {𝐴} ∧ ∃𝑦 𝐵𝑅𝑦) ↔ (𝐵 = 𝐴 ∧ 𝐵 ∈ dom 𝑅)))
6	1, 5	bitrd 281	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝐵 = 𝐴 ∧ 𝐵 ∈ dom 𝑅)))
7		eqelb 38688	. 2 ⊢ ((𝐵 = 𝐴 ∧ 𝐵 ∈ dom 𝑅) ↔ (𝐵 = 𝐴 ∧ 𝐴 ∈ dom 𝑅))
8	6, 7	bitrdi 289	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵 ∈ dom (𝑅 ↾ {𝐴}) ↔ (𝐵 = 𝐴 ∧ 𝐴 ∈ dom 𝑅)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1554  ∃wex 1793   ∈ wcel 2136  {csn 4576   class class class wbr 5094  dom cdm 5640   ↾ cres 5642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728  ax-sep 5240  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-ral 3071  df-rex 3081  df-rab 3409  df-v 3450  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-br 5095  df-opab 5157  df-xp 5646  df-dm 5650  df-res 5652

This theorem is referenced by:  refressn  38980