Theorem op2ndg 7955

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 4816	. . 3 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveqeq2d 6848	. 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦))
3		opeq2 4817	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
4	3	fveq2d 6844	. . 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 3433	. . 3 ⊢ 𝑥 ∈ V
8		vex 3433	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	op2nd 7951	. 2 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
10	2, 6, 9	vtocl2g 3517	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4573  ‘cfv 6498  2^nd c2nd 7941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fv 6506  df-2nd 7943

This theorem is referenced by:  ot2ndg  7957  ot3rdg  7958  br2ndeqg  7965  2ndconst  8051  mposn  8053  curry1  8054  opco2  8074  xpmapenlem  9082  2ndinl  9852  2ndinr  9854  axdc4lem  10377  pinq  10850  addpipq  10860  mulpipq  10863  ordpipq  10865  swrdval  14606  ruclem1  16198  eucalg  16556  qnumdenbi  16714  setsstruct  17146  comffval  17665  oppccofval  17682  funcf2  17835  cofuval2  17854  resfval2  17860  resf2nd  17862  funcres  17863  isnat  17917  fucco  17932  homacd  18008  setcco  18050  catcco  18072  estrcco  18096  xpcco  18149  xpchom2  18152  xpcco2  18153  evlf2  18184  curfval  18189  curf1cl  18194  uncf1  18202  uncf2  18203  hof2fval  18221  yonedalem21  18239  yonedalem22  18244  mvmulfval  22507  imasdsf1olem  24338  ovolicc1  25483  ioombl1lem3  25527  ioombl1lem4  25528  addsqnreup  27406  addsval  27954  mulsval  28101  om2noseqrdg  28296  brcgr  28969  opiedgfv  29076  fsuppcurry1  32797  erlbrd  33324  rlocaddval  33329  rlocmulval  33330  fracerl  33367  sategoelfvb  35601  prv1n  35613  fvtransport  36214  bj-finsumval0  37599  poimirlem17  37958  poimirlem24  37965  poimirlem27  37968  dvhopvadd  41539  dvhopvsca  41548  dvhopaddN  41560  dvhopspN  41561  etransclem44  46706  gpgedgiov  48541  gpgedg2ov  48542  gpgedg2iv  48543  uspgrsprfo  48624  rngccoALTV  48747  ringccoALTV  48781  lmod1zr  48969  func2nd  49553  oppf1st2nd  49606  upfval3  49653  swapf2fval  49740  fucofval  49794  fuco112  49804  fuco21  49811  prcofvala  49852  lanfval  50088  ranfval  50089