MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpeq2 Structured version   Visualization version   GIF version

Theorem xpeq2 5089
Description: Equality theorem for Cartesian product. (Contributed by NM, 5-Jul-1994.)
Assertion
Ref Expression
xpeq2 (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))

Proof of Theorem xpeq2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2687 . . . 4 (𝐴 = 𝐵 → (𝑦𝐴𝑦𝐵))
21anbi2d 739 . . 3 (𝐴 = 𝐵 → ((𝑥𝐶𝑦𝐴) ↔ (𝑥𝐶𝑦𝐵)))
32opabbidv 4678 . 2 (𝐴 = 𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐴)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐵)})
4 df-xp 5080 . 2 (𝐶 × 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐴)}
5 df-xp 5080 . 2 (𝐶 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝑦𝐵)}
63, 4, 53eqtr4g 2680 1 (𝐴 = 𝐵 → (𝐶 × 𝐴) = (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {copab 4672   × cxp 5072
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-opab 4674  df-xp 5080
This theorem is referenced by:  xpeq12  5094  xpeq2i  5096  xpeq2d  5099  xpnz  5512  xpdisj2  5515  dmxpss  5524  rnxpid  5526  xpcan  5529  unixp  5627  fconst5  6425  pmvalg  7813  xpcomeng  7996  unxpdom  8111  marypha1  8284  dfac5lem3  8892  dfac5lem4  8893  hsmexlem8  9190  axdc4uz  12723  hashxp  13161  mamufval  20110  txuni2  21278  txbas  21280  txopn  21315  txrest  21344  txdis  21345  txdis1cn  21348  txtube  21353  txcmplem2  21355  tx1stc  21363  qustgplem  21834  tsmsxplem1  21866  isgrpo  27197  vciOLD  27262  isvclem  27278  issh  27911  hhssablo  27966  hhssnvt  27968  hhsssh  27972  txomap  29680  tpr2rico  29737  elsx  30035  mbfmcst  30099  br2base  30109  dya2iocnrect  30121  sxbrsigalem5  30128  0rrv  30291  dfpo2  31350  elima4  31378  finxpeq1  32852  isbnd3  33212  hdmap1fval  36563  csbresgVD  38611
  Copyright terms: Public domain W3C validator