Theorem op2ndg 7347

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 4553	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveq2d 6357	. . 3 ⊢ (𝑥 = 𝐴 → (2nd ‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝐴, 𝑦⟩))
3	2	eqeq1d 2762	. 2 ⊢ (𝑥 = 𝐴 → ((2nd ‘⟨𝑥, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦))
4		opeq2 4554	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
5	4	fveq2d 6357	. . 3 ⊢ (𝑦 = 𝐵 → (2nd ‘⟨𝐴, 𝑦⟩) = (2nd ‘⟨𝐴, 𝐵⟩))
6		id 22	. . 3 ⊢ (𝑦 = 𝐵 → 𝑦 = 𝐵)
7	5, 6	eqeq12d 2775	. 2 ⊢ (𝑦 = 𝐵 → ((2nd ‘⟨𝐴, 𝑦⟩) = 𝑦 ↔ (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵))
8		vex 3343	. . 3 ⊢ 𝑥 ∈ V
9		vex 3343	. . 3 ⊢ 𝑦 ∈ V
10	8, 9	op2nd 7343	. 2 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
11	3, 7, 10	vtocl2g 3410	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ⟨cop 4327  ‘cfv 6049  2^nd c2nd 7333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-2nd 7335

This theorem is referenced by:  ot2ndg  7349  ot3rdg  7350  br2ndeqg  7357  2ndconst  7435  mpt2sn  7437  curry1  7438  xpmapenlem  8294  2ndinl  8964  2ndinr  8966  axdc4lem  9489  pinq  9961  addpipq  9971  mulpipq  9974  ordpipq  9976  swrdval  13636  ruclem1  15179  eucalg  15522  qnumdenbi  15674  setsstruct  16120  comffval  16580  oppccofval  16597  funcf2  16749  cofuval2  16768  resfval2  16774  resf2nd  16776  funcres  16777  isnat  16828  fucco  16843  homacd  16912  setcco  16954  catcco  16972  estrcco  16991  xpcco  17044  xpchom2  17047  xpcco2  17048  evlf2  17079  curfval  17084  curf1cl  17089  uncf1  17097  uncf2  17098  hof2fval  17116  yonedalem21  17134  yonedalem22  17139  mvmulfval  20570  imasdsf1olem  22399  ovolicc1  23504  ioombl1lem3  23548  ioombl1lem4  23549  brcgr  26000  opiedgfv  26107  fvtransport  32466  bj-elid3  33416  bj-finsumval0  33476  poimirlem17  33757  poimirlem24  33764  poimirlem27  33767  dvhopvadd  36902  dvhopvsca  36911  dvhopaddN  36923  dvhopspN  36924  etransclem44  41016  uspgrsprfo  42284  rngccoALTV  42516  ringccoALTV  42579  lmod1zr  42810