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Theorem fvsn 6935
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 6934 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)
41, 2, 3mp2an 688 1 ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  Vcvv 3492  {csn 4557  cop 4563  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356
This theorem is referenced by:  fvpr1  6944  elixpsn  8489  ac6sfi  8750  dcomex  9857  axdc3lem4  9863  0ram  16344  mdet0fv0  21131  chpmat0d  21370  imasdsf1olem  22910  axlowdimlem8  26662  axlowdimlem11  26665  subfacp1lem2a  32324  subfacp1lem5  32328  cvmliftlem4  32432  frrlem12  33031  finixpnum  34758  poimirlem3  34776  fdc  34901  grposnOLD  35041
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