Theorem op2ndg 7141

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 4377	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveq2d 6162	. . 3 ⊢ (𝑥 = 𝐴 → (2nd ‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝐴, 𝑦⟩))
3	2	eqeq1d 2623	. 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦))
4		opeq2 4378	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
5	4	fveq2d 6162	. . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘⟨𝐴, 𝑦⟩) = (2nd ‘⟨𝐴, 𝐵⟩))
6		id 22	. . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
7	5, 6	eqeq12d 2636	. 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘⟨𝐴, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵))
8		vex 3193	. . 3 ⊢ 𝑥 ∈ V
9		vex 3193	. . 3 ⊢ 𝑦 ∈ V
10	8, 9	op2nd 7137	. 2 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
11	3, 7, 10	vtocl2g 3260	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ⟨cop 4161  ‘cfv 5857  2^nd c2nd 7127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fv 5865  df-2nd 7129

This theorem is referenced by:  ot2ndg  7143  ot3rdg  7144  2ndconst  7226  mpt2sn  7228  curry1  7229  xpmapenlem  8087  axdc4lem  9237  pinq  9709  addpipq  9719  mulpipq  9722  ordpipq  9724  swrdval  13371  ruclem1  14904  eucalg  15243  qnumdenbi  15395  setsstruct  15838  comffval  16299  oppccofval  16316  funcf2  16468  cofuval2  16487  resfval2  16493  resf2nd  16495  funcres  16496  isnat  16547  fucco  16562  homacd  16631  setcco  16673  catcco  16691  estrcco  16710  xpcco  16763  xpchom2  16766  xpcco2  16767  evlf2  16798  curfval  16803  curf1cl  16808  uncf1  16816  uncf2  16817  hof2fval  16835  yonedalem21  16853  yonedalem22  16858  mvmulfval  20288  imasdsf1olem  22118  ovolicc1  23224  ioombl1lem3  23268  ioombl1lem4  23269  brcgr  25714  opiedgfv  25821  fvtransport  31834  bj-elid3  32759  bj-finsumval0  32819  poimirlem17  33097  poimirlem24  33104  poimirlem27  33107  dvhopvadd  35901  dvhopvsca  35910  dvhopaddN  35922  dvhopspN  35923  etransclem44  39832  uspgrsprfo  41074  rngccoALTV  41306  ringccoALTV  41369  lmod1zr  41600