Theorem op2ndg 7934

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.

Step	Hyp	Ref	Expression
1		opeq1 4825	. . 3 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveqeq2d 6830	. 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦))
3		opeq2 4826	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
4	3	fveq2d 6826	. . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘⟨𝐴, 𝑦⟩) = (2nd ‘⟨𝐴, 𝐵⟩))
5		id 22	. . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
6	4, 5	eqeq12d 2747	. 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘⟨𝐴, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵))
7		vex 3440	. . 3 ⊢ 𝑥 ∈ V
8		vex 3440	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	op2nd 7930	. 2 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
10	2, 6, 9	vtocl2g 3529	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ⟨cop 4582  ‘cfv 6481  2^nd c2nd 7920

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-iota 6437  df-fun 6483  df-fv 6489  df-2nd 7922

This theorem is referenced by:  ot2ndg  7936  ot3rdg  7937  br2ndeqg  7944  2ndconst  8031  mposn  8033  curry1  8034  opco2  8054  xpmapenlem  9057  2ndinl  9818  2ndinr  9820  axdc4lem  10343  pinq  10815  addpipq  10825  mulpipq  10828  ordpipq  10830  swrdval  14548  ruclem1  16137  eucalg  16495  qnumdenbi  16652  setsstruct  17084  comffval  17602  oppccofval  17619  funcf2  17772  cofuval2  17791  resfval2  17797  resf2nd  17799  funcres  17800  isnat  17854  fucco  17869  homacd  17945  setcco  17987  catcco  18009  estrcco  18033  xpcco  18086  xpchom2  18089  xpcco2  18090  evlf2  18121  curfval  18126  curf1cl  18131  uncf1  18139  uncf2  18140  hof2fval  18158  yonedalem21  18176  yonedalem22  18181  mvmulfval  22455  imasdsf1olem  24286  ovolicc1  25442  ioombl1lem3  25486  ioombl1lem4  25487  addsqnreup  27379  addsval  27903  mulsval  28046  om2noseqrdg  28232  brcgr  28876  opiedgfv  28983  fsuppcurry1  32702  erlbrd  33225  rlocaddval  33230  rlocmulval  33231  fracerl  33267  sategoelfvb  35451  prv1n  35463  fvtransport  36065  bj-finsumval0  37318  poimirlem17  37676  poimirlem24  37683  poimirlem27  37686  dvhopvadd  41131  dvhopvsca  41140  dvhopaddN  41152  dvhopspN  41153  etransclem44  46315  gpgedgiov  48095  gpgedg2ov  48096  gpgedg2iv  48097  uspgrsprfo  48178  rngccoALTV  48301  ringccoALTV  48335  lmod1zr  48524  func2nd  49109  oppf1st2nd  49162  upfval3  49209  swapf2fval  49296  fucofval  49350  fuco112  49360  fuco21  49367  prcofvala  49408  lanfval  49644  ranfval  49645