Theorem elrnressn 38784

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 38782	. 2 ⊢ (𝐵 ∈ 𝑊 → (𝐵 ∈ ran (𝑅 ↾ {𝐴}) ↔ ∃𝑥 ∈ {𝐴}𝑥𝑅𝐵))
2		breq1 5105	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝐵 ↔ 𝐴𝑅𝐵))
3	2	rexsng 4637	. 2 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝑥𝑅𝐵 ↔ 𝐴𝑅𝐵))
4	1, 3	sylan9bbr 518	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2144  ∃wrex 3088  {csn 4584   class class class wbr 5102  ran crn 5650   ↾ cres 5651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-xp 5655  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661

This theorem is referenced by:  refressn  39037