Theorem op2ndg 7049

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 4334	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveq2d 6092	. . 3 ⊢ (𝑥 = 𝐴 → (2nd ‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝐴, 𝑦⟩))
3	2	eqeq1d 2611	. 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦))
4		opeq2 4335	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
5	4	fveq2d 6092	. . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘⟨𝐴, 𝑦⟩) = (2nd ‘⟨𝐴, 𝐵⟩))
6		id 22	. . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
7	5, 6	eqeq12d 2624	. 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘⟨𝐴, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵))
8		vex 3175	. . 3 ⊢ 𝑥 ∈ V
9		vex 3175	. . 3 ⊢ 𝑦 ∈ V
10	8, 9	op2nd 7045	. 2 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
11	3, 7, 10	vtocl2g 3242	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1474   ∈ wcel 1976  ⟨cop 4130  ‘cfv 5790  2^nd c2nd 7035

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-iota 5754  df-fun 5792  df-fv 5798  df-2nd 7037

This theorem is referenced by:  ot2ndg  7051  ot3rdg  7052  2ndconst  7130  mpt2sn  7132  curry1  7133  xpmapenlem  7989  axdc4lem  9137  pinq  9605  addpipq  9615  mulpipq  9618  ordpipq  9620  swrdval  13215  ruclem1  14745  eucalg  15084  qnumdenbi  15236  comffval  16128  oppccofval  16145  funcf2  16297  cofuval2  16316  resfval2  16322  resf2nd  16324  funcres  16325  isnat  16376  fucco  16391  homacd  16460  setcco  16502  catcco  16520  estrcco  16539  xpcco  16592  xpchom2  16595  xpcco2  16596  evlf2  16627  curfval  16632  curf1cl  16637  uncf1  16645  uncf2  16646  hof2fval  16664  yonedalem21  16682  yonedalem22  16687  mvmulfval  20109  imasdsf1olem  21929  ovolicc1  23008  ioombl1lem3  23052  ioombl1lem4  23053  brcgr  25498  edgopval  25635  nbgraop  25718  nbgraopALT  25719  vcoprneOLD  26600  fvtransport  31115  bj-elid3  32060  bj-finsumval0  32120  poimirlem17  32392  poimirlem24  32399  poimirlem27  32402  dvhopvadd  35196  dvhopvsca  35205  dvhopaddN  35217  dvhopspN  35218  etransclem44  38968  opiedgfv  40235  rngccoALTV  41775  ringccoALTV  41838  lmod1zr  42071