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Theorem fvsng 5302
Description: The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)
Assertion
Ref Expression
fvsng ((A 𝑉 B 𝑊) → ({⟨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 3540 . . . . 5 (𝑎 = A → ⟨𝑎, 𝑏⟩ = ⟨A, 𝑏⟩)
21sneqd 3380 . . . 4 (𝑎 = A → {⟨𝑎, 𝑏⟩} = {⟨A, 𝑏⟩})
3 id 19 . . . 4 (𝑎 = A𝑎 = A)
42, 3fveq12d 5127 . . 3 (𝑎 = A → ({⟨𝑎, 𝑏⟩}‘𝑎) = ({⟨A, 𝑏⟩}‘A))
54eqeq1d 2045 . 2 (𝑎 = A → (({⟨𝑎, 𝑏⟩}‘𝑎) = 𝑏 ↔ ({⟨A, 𝑏⟩}‘A) = 𝑏))
6 opeq2 3541 . . . . 5 (𝑏 = B → ⟨A, 𝑏⟩ = ⟨A, B⟩)
76sneqd 3380 . . . 4 (𝑏 = B → {⟨A, 𝑏⟩} = {⟨A, B⟩})
87fveq1d 5123 . . 3 (𝑏 = B → ({⟨A, 𝑏⟩}‘A) = ({⟨A, B⟩}‘A))
9 id 19 . . 3 (𝑏 = B𝑏 = B)
108, 9eqeq12d 2051 . 2 (𝑏 = B → (({⟨A, 𝑏⟩}‘A) = 𝑏 ↔ ({⟨A, B⟩}‘A) = B))
11 vex 2554 . . 3 𝑎 V
12 vex 2554 . . 3 𝑏 V
1311, 12fvsn 5301 . 2 ({⟨𝑎, 𝑏⟩}‘𝑎) = 𝑏
145, 10, 13vtocl2g 2611 1 ((A 𝑉 B 𝑊) → ({⟨A, B⟩}‘A) = B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  {csn 3367  cop 3370  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853
This theorem is referenced by:  fsnunfv  5306  fvpr1g  5310  fvpr2g  5311  tfr0  5878  fseq1p1m1  8726  1fv  8766
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