MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpoeq12 Structured version   Visualization version   GIF version

Theorem mpoeq12 7482
Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Assertion
Ref Expression
mpoeq12 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝑥𝐴, 𝑦𝐵𝐸) = (𝑥𝐶, 𝑦𝐷𝐸))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦
Allowed substitution hints:   𝐸(𝑥,𝑦)

Proof of Theorem mpoeq12
StepHypRef Expression
1 eqid 2733 . . . . 5 𝐸 = 𝐸
21rgenw 3066 . . . 4 𝑦𝐵 𝐸 = 𝐸
32jctr 526 . . 3 (𝐵 = 𝐷 → (𝐵 = 𝐷 ∧ ∀𝑦𝐵 𝐸 = 𝐸))
43ralrimivw 3151 . 2 (𝐵 = 𝐷 → ∀𝑥𝐴 (𝐵 = 𝐷 ∧ ∀𝑦𝐵 𝐸 = 𝐸))
5 mpoeq123 7481 . 2 ((𝐴 = 𝐶 ∧ ∀𝑥𝐴 (𝐵 = 𝐷 ∧ ∀𝑦𝐵 𝐸 = 𝐸)) → (𝑥𝐴, 𝑦𝐵𝐸) = (𝑥𝐶, 𝑦𝐷𝐸))
64, 5sylan2 594 1 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝑥𝐴, 𝑦𝐵𝐸) = (𝑥𝐶, 𝑦𝐷𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wral 3062  cmpo 7411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-oprab 7413  df-mpo 7414
This theorem is referenced by:  dffi3  9426  cantnfres  9672  xpsval  17516  monpropd  17684  grpsubpropd2  18929  lsmvalx  19507  lsmpropd  19545  psrplusgpropd  21758  d1mat2pmat  22241  txval  23068  cnmptk1p  23189  cnmptk2  23190  xpstopnlem1  23313  rrxmval  24922  madjusmdetlem1  32807  pstmval  32875  qqhval2  32962  funcrngcsetc  46896  funcringcsetc  46933  lmod1zr  47174
  Copyright terms: Public domain W3C validator