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Theorem xpeq1 5093
Description: Equality theorem for Cartesian product. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
xpeq1 (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))

Proof of Theorem xpeq1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2687 . . . 4 (𝐴 = 𝐵 → (𝑥𝐴𝑥𝐵))
21anbi1d 740 . . 3 (𝐴 = 𝐵 → ((𝑥𝐴𝑦𝐶) ↔ (𝑥𝐵𝑦𝐶)))
32opabbidv 4683 . 2 (𝐴 = 𝐵 → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)})
4 df-xp 5085 . 2 (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)}
5 df-xp 5085 . 2 (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)}
63, 4, 53eqtr4g 2680 1 (𝐴 = 𝐵 → (𝐴 × 𝐶) = (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {copab 4677   × cxp 5077
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 4679  df-xp 5085
This theorem is referenced by:  xpeq12  5099  xpeq1i  5100  xpeq1d  5103  opthprc  5132  dmxpid  5310  reseq2  5356  xpnz  5517  xpdisj1  5519  xpcan2  5535  xpima  5540  unixp  5632  unixpid  5634  pmvalg  7820  xpsneng  7997  xpcomeng  8004  xpdom2g  8008  fodomr  8063  unxpdom  8119  xpfi  8183  marypha1lem  8291  cdaval  8944  iundom2g  9314  hashxplem  13168  dmtrclfv  13701  ramcl  15668  efgval  18062  frgpval  18103  frlmval  20024  txuni2  21291  txbas  21293  txopn  21328  txrest  21357  txdis  21358  txdis1cn  21361  tx1stc  21376  tmdgsum  21822  qustgplem  21847  incistruhgr  25887  isgrpo  27221  hhssablo  27990  hhssnvt  27992  hhsssh  27996  txomap  29707  tpr2rico  29764  elsx  30062  br2base  30136  dya2iocnrect  30148  sxbrsigalem5  30155  sibf0  30201  cvmlift2lem13  31040
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