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Theorem fvsng 5664
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 3742 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
21sneqd 3573 . . . 4 (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩})
3 id 19 . . . 4 (𝑎 = 𝐴𝑎 = 𝐴)
42, 3fveq12d 5476 . . 3 (𝑎 = 𝐴 → ({⟨𝑎, 𝑏⟩}‘𝑎) = ({⟨𝐴, 𝑏⟩}‘𝐴))
54eqeq1d 2166 . 2 (𝑎 = 𝐴 → (({⟨𝑎, 𝑏⟩}‘𝑎) = 𝑏 ↔ ({⟨𝐴, 𝑏⟩}‘𝐴) = 𝑏))
6 opeq2 3743 . . . . 5 (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
76sneqd 3573 . . . 4 (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩})
87fveq1d 5471 . . 3 (𝑏 = 𝐵 → ({⟨𝐴, 𝑏⟩}‘𝐴) = ({⟨𝐴, 𝐵⟩}‘𝐴))
9 id 19 . . 3 (𝑏 = 𝐵𝑏 = 𝐵)
108, 9eqeq12d 2172 . 2 (𝑏 = 𝐵 → (({⟨𝐴, 𝑏⟩}‘𝐴) = 𝑏 ↔ ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵))
11 vex 2715 . . 3 𝑎 ∈ V
12 vex 2715 . . 3 𝑏 ∈ V
1311, 12fvsn 5663 . 2 ({⟨𝑎, 𝑏⟩}‘𝑎) = 𝑏
145, 10, 13vtocl2g 2776 1 ((𝐴𝑉𝐵𝑊) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1335  wcel 2128  {csn 3560  cop 3563  cfv 5171
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-iota 5136  df-fun 5173  df-fv 5179
This theorem is referenced by:  fsnunfv  5669  fvpr1g  5674  fvpr2g  5675  tfr0dm  6270  fseq1p1m1  9997  1fv  10042  sumsnf  11310  prodsnf  11493  setsslid  12282
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