Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elrnressn Structured version   Visualization version   GIF version

Theorem elrnressn 37654
Description: Element of the range of a restriction to a singleton. (Contributed by Peter Mazsa, 12-Jun-2024.)
Assertion
Ref Expression
elrnressn ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝐵))

Proof of Theorem elrnressn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elrnres 37652 . 2 (𝐵𝑊 → (𝐵 ∈ ran (𝑅 ↾ {𝐴}) ↔ ∃𝑥 ∈ {𝐴}𝑥𝑅𝐵))
2 breq1 5144 . . 3 (𝑥 = 𝐴 → (𝑥𝑅𝐵𝐴𝑅𝐵))
32rexsng 4673 . 2 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝑥𝑅𝐵𝐴𝑅𝐵))
41, 3sylan9bbr 510 1 ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ ran (𝑅 ↾ {𝐴}) ↔ 𝐴𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2098  wrex 3064  {csn 4623   class class class wbr 5141  ran crn 5670  cres 5671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681
This theorem is referenced by:  refressn  37826
  Copyright terms: Public domain W3C validator