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Theorem op2ndg 6049
Description: Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.)
Assertion
Ref Expression
op2ndg ((𝐴𝑉𝐵𝑊) → (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 3705 . . . 4 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
21fveq2d 5425 . . 3 (𝑥 = 𝐴 → (2nd ‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝐴, 𝑦⟩))
32eqeq1d 2148 . 2 (𝑥 = 𝐴 → ((2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦))
4 opeq2 3706 . . . 4 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
54fveq2d 5425 . . 3 (𝑦 = 𝐵 → (2nd ‘⟨𝐴, 𝑦⟩) = (2nd ‘⟨𝐴, 𝐵⟩))
6 id 19 . . 3 (𝑦 = 𝐵𝑦 = 𝐵)
75, 6eqeq12d 2154 . 2 (𝑦 = 𝐵 → ((2nd ‘⟨𝐴, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵))
8 vex 2689 . . 3 𝑥 ∈ V
9 vex 2689 . . 3 𝑦 ∈ V
108, 9op2nd 6045 . 2 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
113, 7, 10vtocl2g 2750 1 ((𝐴𝑉𝐵𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  cop 3530  cfv 5123  2nd c2nd 6037
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-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-2nd 6039
This theorem is referenced by:  ot2ndg  6051  ot3rdgg  6052  2ndconst  6119  xpmapenlem  6743  2ndinl  6960  2ndinr  6962  mulpipq  7187  suplocexprlem2b  7529  aprcl  8415  frec2uzrdg  10189  frecuzrdgsuc  10194  eucalglt  11745  eucalg  11747  qredeu  11785  sqpweven  11860  2sqpwodd  11861  qnumdenbi  11877  upxp  12451  uptx  12453  txmetcnp  12697
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