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Theorem fvsn 7201
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 7200 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)
41, 2, 3mp2an 692 1 ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631  cop 4637  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571
This theorem is referenced by:  frrlem12  8321  elixpsn  8976  ac6sfi  9318  dcomex  10485  axdc3lem4  10491  0ram  17054  mdet0fv0  22616  chpmat0d  22856  imasdsf1olem  24399  axlowdimlem8  28979  axlowdimlem11  28982  subfacp1lem2a  35165  subfacp1lem5  35169  cvmliftlem4  35273  finixpnum  37592  poimirlem3  37610  fdc  37732  grposnOLD  37869  1arymaptfo  48493  mndtchom  48893  mndtcco  48894
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