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Theorem fvsn 6590
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 6082 . 2 Fun {⟨𝐴, 𝐵⟩}
4 opex 5060 . . 3 𝐴, 𝐵⟩ ∈ V
54snid 4347 . 2 𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩}
6 funopfv 6376 . 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 1631  wcel 2145  Vcvv 3351  {csn 4316  cop 4322  Fun wfun 6025  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039
This theorem is referenced by:  fvsng  6591  fvsnun1  6592  fvpr1  6600  elixpsn  8101  ac6sfi  8360  dcomex  9471  axdc3lem4  9477  0ram  15931  mdet0fv0  20618  chpmat0d  20859  imasdsf1olem  22398  axlowdimlem8  26050  axlowdimlem11  26053  subfacp1lem2a  31500  subfacp1lem5  31504  cvmliftlem4  31608  finixpnum  33727  poimirlem3  33745  fdc  33873  grposnOLD  34013
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