Theorem op2ndg 7817

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 4801	. . 3 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveqeq2d 6764	. 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦))
3		opeq2 4802	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
4	3	fveq2d 6760	. . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘⟨𝐴, 𝑦⟩) = (2nd ‘⟨𝐴, 𝐵⟩))
5		id 22	. . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
6	4, 5	eqeq12d 2754	. 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘⟨𝐴, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵))
7		vex 3426	. . 3 ⊢ 𝑥 ∈ V
8		vex 3426	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	op2nd 7813	. 2 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
10	2, 6, 9	vtocl2g 3500	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ⟨cop 4564  ‘cfv 6418  2^nd c2nd 7803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fv 6426  df-2nd 7805

This theorem is referenced by:  ot2ndg  7819  ot3rdg  7820  br2ndeqg  7827  2ndconst  7912  mposn  7914  curry1  7915  opco2  7936  xpmapenlem  8880  2ndinl  9617  2ndinr  9619  axdc4lem  10142  pinq  10614  addpipq  10624  mulpipq  10627  ordpipq  10629  swrdval  14284  ruclem1  15868  eucalg  16220  qnumdenbi  16376  setsstruct  16805  comffval  17325  oppccofval  17343  funcf2  17499  cofuval2  17518  resfval2  17524  resf2nd  17526  funcres  17527  isnat  17579  fucco  17596  homacd  17672  setcco  17714  catcco  17736  estrcco  17762  xpcco  17816  xpchom2  17819  xpcco2  17820  evlf2  17852  curfval  17857  curf1cl  17862  uncf1  17870  uncf2  17871  hof2fval  17889  yonedalem21  17907  yonedalem22  17912  mvmulfval  21599  imasdsf1olem  23434  ovolicc1  24585  ioombl1lem3  24629  ioombl1lem4  24630  addsqnreup  26496  brcgr  27171  opiedgfv  27280  fsuppcurry1  30962  sategoelfvb  33281  prv1n  33293  addsval  34053  fvtransport  34261  bj-finsumval0  35383  poimirlem17  35721  poimirlem24  35728  poimirlem27  35731  dvhopvadd  39034  dvhopvsca  39043  dvhopaddN  39055  dvhopspN  39056  etransclem44  43709  uspgrsprfo  45198  rngccoALTV  45434  ringccoALTV  45497  lmod1zr  45722