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Theorem fvsn 5583
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 5141 . 2 Fun {⟨𝐴, 𝐵⟩}
41, 2opex 4121 . . 3 𝐴, 𝐵⟩ ∈ V
54snid 3526 . 2 𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩}
6 funopfv 5429 . 2 (Fun {⟨𝐴, 𝐵⟩} → (⟨𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩} → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵))
73, 5, 6mp2 16 1 ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵
Colors of variables: wff set class
Syntax hints:   = wceq 1316  wcel 1465  Vcvv 2660  {csn 3497  cop 3500  Fun wfun 5087  cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101
This theorem is referenced by:  fvsng  5584  fvsnun1  5585  fvpr1  5592  elixpsn  6597  mapsnen  6673  ac6sfi  6760
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