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Theorem mpoeq12 7423
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 2738 . . . . 5 𝐸 = 𝐸
21rgenw 3067 . . . 4 𝑦𝐵 𝐸 = 𝐸
32jctr 526 . . 3 (𝐵 = 𝐷 → (𝐵 = 𝐷 ∧ ∀𝑦𝐵 𝐸 = 𝐸))
43ralrimivw 3146 . 2 (𝐵 = 𝐷 → ∀𝑥𝐴 (𝐵 = 𝐷 ∧ ∀𝑦𝐵 𝐸 = 𝐸))
5 mpoeq123 7422 . 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 3063  cmpo 7352
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 2709
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 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ral 3064  df-oprab 7354  df-mpo 7355
This theorem is referenced by:  dffi3  9301  cantnfres  9547  xpsval  17388  monpropd  17556  grpsubpropd2  18788  lsmvalx  19356  lsmpropd  19394  psrplusgpropd  21535  d1mat2pmat  22016  txval  22843  cnmptk1p  22964  cnmptk2  22965  xpstopnlem1  23088  rrxmval  24697  madjusmdetlem1  32188  pstmval  32256  qqhval2  32343  funcrngcsetc  46087  funcringcsetc  46124  lmod1zr  46365
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