Theorem op2ndg 7990

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 4872	. . 3 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveqeq2d 6898	. 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦))
3		opeq2 4873	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
4	3	fveq2d 6894	. . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘⟨𝐴, 𝑦⟩) = (2nd ‘⟨𝐴, 𝐵⟩))
5		id 22	. . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
6	4, 5	eqeq12d 2746	. 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘⟨𝐴, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵))
7		vex 3476	. . 3 ⊢ 𝑥 ∈ V
8		vex 3476	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	op2nd 7986	. 2 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
10	2, 6, 9	vtocl2g 3562	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ⟨cop 4633  ‘cfv 6542  2^nd c2nd 7976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fv 6550  df-2nd 7978

This theorem is referenced by:  ot2ndg  7992  ot3rdg  7993  br2ndeqg  8000  2ndconst  8089  mposn  8091  curry1  8092  opco2  8112  xpmapenlem  9146  2ndinl  9925  2ndinr  9927  axdc4lem  10452  pinq  10924  addpipq  10934  mulpipq  10937  ordpipq  10939  swrdval  14597  ruclem1  16178  eucalg  16528  qnumdenbi  16684  setsstruct  17113  comffval  17647  oppccofval  17665  funcf2  17822  cofuval2  17841  resfval2  17847  resf2nd  17849  funcres  17850  isnat  17902  fucco  17919  homacd  17995  setcco  18037  catcco  18059  estrcco  18085  xpcco  18139  xpchom2  18142  xpcco2  18143  evlf2  18175  curfval  18180  curf1cl  18185  uncf1  18193  uncf2  18194  hof2fval  18212  yonedalem21  18230  yonedalem22  18235  mvmulfval  22264  imasdsf1olem  24099  ovolicc1  25265  ioombl1lem3  25309  ioombl1lem4  25310  addsqnreup  27182  addsval  27684  mulsval  27804  brcgr  28425  opiedgfv  28534  fsuppcurry1  32217  sategoelfvb  34708  prv1n  34720  fvtransport  35308  bj-finsumval0  36469  poimirlem17  36808  poimirlem24  36815  poimirlem27  36818  dvhopvadd  40267  dvhopvsca  40276  dvhopaddN  40288  dvhopspN  40289  etransclem44  45292  uspgrsprfo  46824  rngccoALTV  46974  ringccoALTV  47037  lmod1zr  47261