Theorem op2ndg 8043

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 4897	. . 3 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveqeq2d 6928	. 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦))
3		opeq2 4898	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
4	3	fveq2d 6924	. . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘⟨𝐴, 𝑦⟩) = (2nd ‘⟨𝐴, 𝐵⟩))
5		id 22	. . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
6	4, 5	eqeq12d 2756	. 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘⟨𝐴, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵))
7		vex 3492	. . 3 ⊢ 𝑥 ∈ V
8		vex 3492	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	op2nd 8039	. 2 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
10	2, 6, 9	vtocl2g 3586	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ⟨cop 4654  ‘cfv 6573  2^nd c2nd 8029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-2nd 8031

This theorem is referenced by:  ot2ndg  8045  ot3rdg  8046  br2ndeqg  8053  2ndconst  8142  mposn  8144  curry1  8145  opco2  8165  xpmapenlem  9210  2ndinl  9997  2ndinr  9999  axdc4lem  10524  pinq  10996  addpipq  11006  mulpipq  11009  ordpipq  11011  swrdval  14691  ruclem1  16279  eucalg  16634  qnumdenbi  16791  setsstruct  17223  comffval  17757  oppccofval  17775  funcf2  17932  cofuval2  17951  resfval2  17957  resf2nd  17959  funcres  17960  isnat  18015  fucco  18032  homacd  18108  setcco  18150  catcco  18172  estrcco  18198  xpcco  18252  xpchom2  18255  xpcco2  18256  evlf2  18288  curfval  18293  curf1cl  18298  uncf1  18306  uncf2  18307  hof2fval  18325  yonedalem21  18343  yonedalem22  18348  mvmulfval  22569  imasdsf1olem  24404  ovolicc1  25570  ioombl1lem3  25614  ioombl1lem4  25615  addsqnreup  27505  addsval  28013  mulsval  28153  om2noseqrdg  28328  brcgr  28933  opiedgfv  29042  fsuppcurry1  32739  erlbrd  33235  rlocaddval  33240  rlocmulval  33241  fracerl  33273  sategoelfvb  35387  prv1n  35399  fvtransport  35996  bj-finsumval0  37251  poimirlem17  37597  poimirlem24  37604  poimirlem27  37607  dvhopvadd  41050  dvhopvsca  41059  dvhopaddN  41071  dvhopspN  41072  etransclem44  46199  uspgrsprfo  47871  rngccoALTV  47994  ringccoALTV  48028  lmod1zr  48222