Theorem elrnressn 38296

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

Assertion

Ref	Expression
elrnressn	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝐵))

Proof of Theorem elrnressn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elrnres 38294	. 2 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ ran (𝑅 ↾ {𝐴}) ↔ ∃𝑥 ∈ {𝐴}𝑥𝑅𝐵))
2		breq1 5127	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝐵 ↔ 𝐴𝑅𝐵))
3	2	rexsng 4657	. 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝑥𝑅𝐵 ↔ 𝐴𝑅𝐵))
4	1, 3	sylan9bbr 510	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  ∃wrex 3061  {csn 4606   class class class wbr 5124  ran crn 5660   ↾ cres 5661

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671

This theorem is referenced by:  refressn  38466