Theorem op2ndg 7704

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 4805	. . 3 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveqeq2d 6680	. 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦))
3		opeq2 4806	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
4	3	fveq2d 6676	. . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘⟨𝐴, 𝑦⟩) = (2nd ‘⟨𝐴, 𝐵⟩))
5		id 22	. . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
6	4, 5	eqeq12d 2839	. 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘⟨𝐴, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵))
7		vex 3499	. . 3 ⊢ 𝑥 ∈ V
8		vex 3499	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	op2nd 7700	. 2 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
10	2, 6, 9	vtocl2g 3574	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ⟨cop 4575  ‘cfv 6357  2^nd c2nd 7690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fv 6365  df-2nd 7692

This theorem is referenced by:  ot2ndg  7706  ot3rdg  7707  br2ndeqg  7714  2ndconst  7798  mposn  7800  curry1  7801  xpmapenlem  8686  2ndinl  9359  2ndinr  9361  axdc4lem  9879  pinq  10351  addpipq  10361  mulpipq  10364  ordpipq  10366  swrdval  14007  ruclem1  15586  eucalg  15933  qnumdenbi  16086  setsstruct  16525  comffval  16971  oppccofval  16988  funcf2  17140  cofuval2  17159  resfval2  17165  resf2nd  17167  funcres  17168  isnat  17219  fucco  17234  homacd  17303  setcco  17345  catcco  17363  estrcco  17382  xpcco  17435  xpchom2  17438  xpcco2  17439  evlf2  17470  curfval  17475  curf1cl  17480  uncf1  17488  uncf2  17489  hof2fval  17507  yonedalem21  17525  yonedalem22  17530  mvmulfval  21153  imasdsf1olem  22985  ovolicc1  24119  ioombl1lem3  24163  ioombl1lem4  24164  addsqnreup  26021  brcgr  26688  opiedgfv  26794  fsuppcurry1  30463  sategoelfvb  32668  prv1n  32680  fvtransport  33495  bj-finsumval0  34569  poimirlem17  34911  poimirlem24  34918  poimirlem27  34921  dvhopvadd  38231  dvhopvsca  38240  dvhopaddN  38252  dvhopspN  38253  etransclem44  42570  uspgrsprfo  44030  rngccoALTV  44266  ringccoALTV  44329  lmod1zr  44555