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Theorem fvsng 5609
 Description: The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)
Assertion
Ref Expression
fvsng ((𝐴𝑉𝐵𝑊) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)

Proof of Theorem fvsng
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3700 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
21sneqd 3535 . . . 4 (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩})
3 id 19 . . . 4 (𝑎 = 𝐴𝑎 = 𝐴)
42, 3fveq12d 5421 . . 3 (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}‘𝑎) = ({⟨𝐴, 𝑏⟩}‘𝐴))
54eqeq1d 2146 . 2 (𝑎 = 𝐴 → (({⟨𝑎, 𝑏⟩}‘𝑎) = 𝑏 ↔ ({⟨𝐴, 𝑏⟩}‘𝐴) = 𝑏))
6 opeq2 3701 . . . . 5 (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
76sneqd 3535 . . . 4 (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩})
87fveq1d 5416 . . 3 (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}‘𝐴) = ({⟨𝐴, 𝐵⟩}‘𝐴))
9 id 19 . . 3 (𝑏 = 𝐵𝑏 = 𝐵)
108, 9eqeq12d 2152 . 2 (𝑏 = 𝐵 → (({⟨𝐴, 𝑏⟩}‘𝐴) = 𝑏 ↔ ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵))
11 vex 2684 . . 3 𝑎 ∈ V
12 vex 2684 . . 3 𝑏 ∈ V
1311, 12fvsn 5608 . 2 ({⟨𝑎, 𝑏⟩}‘𝑎) = 𝑏
145, 10, 13vtocl2g 2745 1 ((𝐴𝑉𝐵𝑊) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  {csn 3522  ⟨cop 3525  ‘cfv 5118 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126 This theorem is referenced by:  fsnunfv  5614  fvpr1g  5619  fvpr2g  5620  tfr0dm  6212  fseq1p1m1  9867  1fv  9909  sumsnf  11171  setsslid  11998
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