Theorem op2ndg 7946

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 4829	. . 3 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveqeq2d 6842	. 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦))
3		opeq2 4830	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
4	3	fveq2d 6838	. . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘⟨𝐴, 𝑦⟩) = (2nd ‘⟨𝐴, 𝐵⟩))
5		id 22	. . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
6	4, 5	eqeq12d 2752	. 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘⟨𝐴, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵))
7		vex 3444	. . 3 ⊢ 𝑥 ∈ V
8		vex 3444	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	op2nd 7942	. 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 2113  ⟨cop 4586  ‘cfv 6492  2^nd c2nd 7932

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-2nd 7934

This theorem is referenced by:  ot2ndg  7948  ot3rdg  7949  br2ndeqg  7956  2ndconst  8043  mposn  8045  curry1  8046  opco2  8066  xpmapenlem  9072  2ndinl  9840  2ndinr  9842  axdc4lem  10365  pinq  10838  addpipq  10848  mulpipq  10851  ordpipq  10853  swrdval  14567  ruclem1  16156  eucalg  16514  qnumdenbi  16671  setsstruct  17103  comffval  17622  oppccofval  17639  funcf2  17792  cofuval2  17811  resfval2  17817  resf2nd  17819  funcres  17820  isnat  17874  fucco  17889  homacd  17965  setcco  18007  catcco  18029  estrcco  18053  xpcco  18106  xpchom2  18109  xpcco2  18110  evlf2  18141  curfval  18146  curf1cl  18151  uncf1  18159  uncf2  18160  hof2fval  18178  yonedalem21  18196  yonedalem22  18201  mvmulfval  22486  imasdsf1olem  24317  ovolicc1  25473  ioombl1lem3  25517  ioombl1lem4  25518  addsqnreup  27410  addsval  27958  mulsval  28105  om2noseqrdg  28300  brcgr  28973  opiedgfv  29080  fsuppcurry1  32803  erlbrd  33345  rlocaddval  33350  rlocmulval  33351  fracerl  33388  sategoelfvb  35613  prv1n  35625  fvtransport  36226  bj-finsumval0  37486  poimirlem17  37834  poimirlem24  37841  poimirlem27  37844  dvhopvadd  41349  dvhopvsca  41358  dvhopaddN  41370  dvhopspN  41371  etransclem44  46518  gpgedgiov  48307  gpgedg2ov  48308  gpgedg2iv  48309  uspgrsprfo  48390  rngccoALTV  48513  ringccoALTV  48547  lmod1zr  48735  func2nd  49319  oppf1st2nd  49372  upfval3  49419  swapf2fval  49506  fucofval  49560  fuco112  49570  fuco21  49577  prcofvala  49618  lanfval  49854  ranfval  49855