Theorem op2ndg 7338

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 4545	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveq2d 6348	. . 3 ⊢ (𝑥 = 𝐴 → (2nd ‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝐴, 𝑦⟩))
3	2	eqeq1d 2754	. 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦))
4		opeq2 4546	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
5	4	fveq2d 6348	. . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘⟨𝐴, 𝑦⟩) = (2nd ‘⟨𝐴, 𝐵⟩))
6		id 22	. . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
7	5, 6	eqeq12d 2767	. 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘⟨𝐴, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵))
8		vex 3335	. . 3 ⊢ 𝑥 ∈ V
9		vex 3335	. . 3 ⊢ 𝑦 ∈ V
10	8, 9	op2nd 7334	. 2 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
11	3, 7, 10	vtocl2g 3402	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1624   ∈ wcel 2131  ⟨cop 4319  ‘cfv 6041  2^nd c2nd 7324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-iota 6004  df-fun 6043  df-fv 6049  df-2nd 7326

This theorem is referenced by:  ot2ndg  7340  ot3rdg  7341  br2ndeqg  7348  2ndconst  7426  mpt2sn  7428  curry1  7429  xpmapenlem  8284  axdc4lem  9461  pinq  9933  addpipq  9943  mulpipq  9946  ordpipq  9948  swrdval  13608  ruclem1  15151  eucalg  15494  qnumdenbi  15646  setsstruct  16092  comffval  16552  oppccofval  16569  funcf2  16721  cofuval2  16740  resfval2  16746  resf2nd  16748  funcres  16749  isnat  16800  fucco  16815  homacd  16884  setcco  16926  catcco  16944  estrcco  16963  xpcco  17016  xpchom2  17019  xpcco2  17020  evlf2  17051  curfval  17056  curf1cl  17061  uncf1  17069  uncf2  17070  hof2fval  17088  yonedalem21  17106  yonedalem22  17111  mvmulfval  20542  imasdsf1olem  22371  ovolicc1  23476  ioombl1lem3  23520  ioombl1lem4  23521  brcgr  25971  opiedgfv  26078  fvtransport  32437  bj-elid3  33390  bj-finsumval0  33450  poimirlem17  33731  poimirlem24  33738  poimirlem27  33741  dvhopvadd  36876  dvhopvsca  36885  dvhopaddN  36897  dvhopspN  36898  etransclem44  40990  uspgrsprfo  42258  rngccoALTV  42490  ringccoALTV  42553  lmod1zr  42784