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Theorem fvsn 5706
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 5259 . 2 Fun {⟨𝐴, 𝐵⟩}
41, 2opex 4225 . . 3 𝐴, 𝐵⟩ ∈ V
54snid 3622 . 2 𝐴, 𝐵⟩ ∈ {⟨𝐴, 𝐵⟩}
6 funopfv 5550 . 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 1353  wcel 2148  Vcvv 2737  {csn 3591  cop 3594  Fun wfun 5205  cfv 5211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fv 5219
This theorem is referenced by:  fvsng  5707  fvsnun1  5708  fvpr1  5715  elixpsn  6728  mapsnen  6804  ac6sfi  6891
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