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Theorem fvsn 6939
 Description: The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.) (Proof shortened by BJ, 25-Feb-2023.)
Hypotheses
Ref Expression
fvsn.1 𝐴 ∈ V
fvsn.2 𝐵 ∈ V
Assertion
Ref Expression
fvsn ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵

Proof of Theorem fvsn
StepHypRef Expression
1 fvsn.1 . 2 𝐴 ∈ V
2 fvsn.2 . 2 𝐵 ∈ V
3 fvsng 6938 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)
41, 2, 3mp2an 691 1 ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  Vcvv 3409  {csn 4525  ⟨cop 4531  ‘cfv 6339 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-iota 6298  df-fun 6341  df-fv 6347 This theorem is referenced by:  fvpr1  6948  elixpsn  8524  ac6sfi  8800  dcomex  9912  axdc3lem4  9918  0ram  16416  mdet0fv0  21299  chpmat0d  21539  imasdsf1olem  23080  axlowdimlem8  26847  axlowdimlem11  26850  subfacp1lem2a  32662  subfacp1lem5  32666  cvmliftlem4  32770  frrlem12  33400  finixpnum  35348  poimirlem3  35366  fdc  35489  grposnOLD  35626  1arymaptfo  45450
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