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Theorem fvsn 6431
Description: The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.)
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 . . 3 𝐴 ∈ V
2 fvsn.2 . . 3 𝐵 ∈ V
31, 2funsn 5927 . 2 Fun {⟨𝐴, 𝐵⟩}
4 opex 4923 . . 3 𝐴, 𝐵⟩ ∈ V
54snid 4199 . 2 𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩}
6 funopfv 6222 . 2 (Fun {⟨𝐴, 𝐵⟩} → (⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩} → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵))
73, 5, 6mp2 9 1 ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1481  wcel 1988  Vcvv 3195  {csn 4168  cop 4174  Fun wfun 5870  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884
This theorem is referenced by:  fvsng  6432  fvsnun1  6433  fvpr1  6441  elixpsn  7932  mapsnen  8020  ac6sfi  8189  dcomex  9254  axdc3lem4  9260  0ram  15705  mdet0fv0  20381  chpmat0d  20620  imasdsf1olem  22159  axlowdimlem8  25810  axlowdimlem11  25813  subfacp1lem2a  31136  subfacp1lem5  31140  cvmliftlem4  31244  finixpnum  33365  poimirlem3  33383  fdc  33512  grposnOLD  33652
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