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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 1540  wcel 2108  Vcvv 3480  {csn 4626  cop 4632  cfv 6561
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569
This theorem is referenced by:  frrlem12  8322  elixpsn  8977  ac6sfi  9320  dcomex  10487  axdc3lem4  10493  0ram  17058  mdet0fv0  22600  chpmat0d  22840  imasdsf1olem  24383  axlowdimlem8  28964  axlowdimlem11  28967  subfacp1lem2a  35185  subfacp1lem5  35189  cvmliftlem4  35293  finixpnum  37612  poimirlem3  37630  fdc  37752  grposnOLD  37889  1arymaptfo  48564  mndtchom  49181  mndtcco  49182
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