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Theorem mpoeq12 7484
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 2730 . . . . 5 𝐸 = 𝐸
21rgenw 3063 . . . 4 𝑦𝐵 𝐸 = 𝐸
32jctr 523 . . 3 (𝐵 = 𝐷 → (𝐵 = 𝐷 ∧ ∀𝑦𝐵 𝐸 = 𝐸))
43ralrimivw 3148 . 2 (𝐵 = 𝐷 → ∀𝑥𝐴 (𝐵 = 𝐷 ∧ ∀𝑦𝐵 𝐸 = 𝐸))
5 mpoeq123 7483 . 2 ((𝐴 = 𝐶 ∧ ∀𝑥𝐴 (𝐵 = 𝐷 ∧ ∀𝑦𝐵 𝐸 = 𝐸)) → (𝑥𝐴, 𝑦𝐵𝐸) = (𝑥𝐶, 𝑦𝐷𝐸))
64, 5sylan2 591 1 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝑥𝐴, 𝑦𝐵𝐸) = (𝑥𝐶, 𝑦𝐷𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wral 3059  cmpo 7413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-oprab 7415  df-mpo 7416
This theorem is referenced by:  dffi3  9428  cantnfres  9674  xpsval  17520  monpropd  17688  grpsubpropd2  18965  lsmvalx  19548  lsmpropd  19586  psrplusgpropd  21978  d1mat2pmat  22461  txval  23288  cnmptk1p  23409  cnmptk2  23410  xpstopnlem1  23533  rrxmval  25153  madjusmdetlem1  33105  pstmval  33173  qqhval2  33260  funcrngcsetc  46984  funcringcsetc  47021  lmod1zr  47261
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