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Theorem fvsn 6675
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 6153 . 2 Fun {⟨𝐴, 𝐵⟩}
4 opex 5123 . . 3 𝐴, 𝐵⟩ ∈ V
54snid 4400 . 2 𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩}
6 funopfv 6459 . 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 1653  wcel 2157  Vcvv 3385  {csn 4368  cop 4374  Fun wfun 6095  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109
This theorem is referenced by:  fvsng  6676  fvsnun1  6677  fvpr1  6685  elixpsn  8187  ac6sfi  8446  dcomex  9557  axdc3lem4  9563  0ram  16057  mdet0fv0  20726  chpmat0d  20967  imasdsf1olem  22506  axlowdimlem8  26186  axlowdimlem11  26189  subfacp1lem2a  31679  subfacp1lem5  31683  cvmliftlem4  31787  finixpnum  33883  poimirlem3  33901  fdc  34028  grposnOLD  34168
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