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Theorem op2ndg 5720
Description: Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
Assertion
Ref Expression
op2ndg ((A 𝑉 B 𝑊) → (2nd ‘⟨A, B⟩) = B)

Proof of Theorem op2ndg
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3540 . . . 4 (x = A → ⟨x, y⟩ = ⟨A, y⟩)
21fveq2d 5125 . . 3 (x = A → (2nd ‘⟨x, y⟩) = (2nd ‘⟨A, y⟩))
32eqeq1d 2045 . 2 (x = A → ((2nd ‘⟨x, y⟩) = y ↔ (2nd ‘⟨A, y⟩) = y))
4 opeq2 3541 . . . 4 (y = B → ⟨A, y⟩ = ⟨A, B⟩)
54fveq2d 5125 . . 3 (y = B → (2nd ‘⟨A, y⟩) = (2nd ‘⟨A, B⟩))
6 id 19 . . 3 (y = By = B)
75, 6eqeq12d 2051 . 2 (y = B → ((2nd ‘⟨A, y⟩) = y ↔ (2nd ‘⟨A, B⟩) = B))
8 vex 2554 . . 3 x V
9 vex 2554 . . 3 y V
108, 9op2nd 5716 . 2 (2nd ‘⟨x, y⟩) = y
113, 7, 10vtocl2g 2611 1 ((A 𝑉 B 𝑊) → (2nd ‘⟨A, B⟩) = B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  cop 3370  cfv 4845  2nd c2nd 5708
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-13 1401  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  ax-un 4136
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-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-2nd 5710
This theorem is referenced by:  ot2ndg  5722  ot3rdgg  5723  2ndconst  5785  mulpipq  6356  frec2uzrdg  8876  frecuzrdgsuc  8882
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