Theorem op2ndg 8041

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 4843	. . 3 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveqeq2d 6919	. 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦))
3		opeq2 4844	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
4	3	fveq2d 6915	. . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘⟨𝐴, 𝑦⟩) = (2nd ‘⟨𝐴, 𝐵⟩))
5		id 23	. . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
6	4, 5	eqeq12d 2779	. 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘⟨𝐴, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵))
7		vex 3458	. . 3 ⊢ 𝑥 ∈ V
8		vex 3458	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	op2nd 8037	. 2 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
10	2, 6, 9	vtocl2g 3537	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ⟨cop 4838  ‘cfv 6561  2^nd c2nd 8027

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-iota 6515  df-fun 6563  df-fv 6569  df-2nd 8029

This theorem is used by:  ot2ndg  8043  ot3rdg  8044  br2ndeqg  8051  2ndconst  8142  mposn  8144  curry1  8145  opco2  8165  xpmapenlem  9194  2ndinl  10039  2ndinr  10041  axdc4lem  10563  pinq  11042  addpipq  11052  mulpipq  11055  ordpipq  11057  swrdval  14822  ruclem1  16431  eucalg  16794  qnumdenbi  16952  setsstruct  17386  comffval  17905  oppccofval  17922  funcf2  18075  cofuval2  18094  resfval2  18100  resf2nd  18102  funcres  18103  isnat  18157  fucco  18172  homacd  18248  setcco  18290  catcco  18312  estrcco  18336  xpcco  18389  xpchom2  18392  xpcco2  18393  evlf2  18424  curfval  18429  curf1cl  18434  uncf1  18442  uncf2  18443  hof2fval  18461  yonedalem21  18479  yonedalem22  18484  mvmulfval  22890  imasdsf1olem  24725  ovolicc1  25870  ioombl1lem3  25914  ioombl1lem4  25915  addsqnreup  27803  addsval  28381  mulsval  28528  om2noseqrdg  28723  brcgr  29511  opiedgfv  29618  fsuppcurry1  33389  erlbrd  33900  erld2  33903  rlocaddval  33906  rlocmulval  33907  fracerl  33952  brlexod  36012  brr0d  36014  sategoelfvb  36364  prv1n  36376  fvtransport  36988  bj-finsumval0  38411  poimirlem17  38760  poimirlem24  38767  poimirlem27  38770  dvhopvadd  42353  dvhopvsca  42362  dvhopaddN  42374  dvhopspN  42375  etransclem44  47503  gpgedgiov  49378  gpgedg2ov  49379  gpgedg2iv  49380  uspgrsprfo  49461  rngccoALTV  49583  ringccoALTV  49617  lmod1zr  49820  func2nd  50407  ixpovf  50412  oppf1st2nd  50462  upfval3  50509  swapf2fval  50602  fucofval  50656  fuco112  50666  fuco21  50673  prcofvala  50727  lanfval  51011  ranfval  51012