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Theorem fconstmpo 7551
Description: Representation of a constant operation using the mapping operation. (Contributed by SO, 11-Jul-2018.)
Assertion
Ref Expression
fconstmpo ((𝐴 × 𝐵) × {𝐶}) = (𝑥𝐴, 𝑦𝐵𝐶)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem fconstmpo
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 fconstmpt 5746 . 2 ((𝐴 × 𝐵) × {𝐶}) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶)
2 eqidd 2737 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐶)
32mpompt 7548 . 2 (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐶)
41, 3eqtri 2764 1 ((𝐴 × 𝐵) × {𝐶}) = (𝑥𝐴, 𝑦𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  {csn 4625  cop 4631  cmpt 5224   × cxp 5682  cmpo 7434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-iun 4992  df-opab 5205  df-mpt 5225  df-xp 5690  df-rel 5691  df-oprab 7436  df-mpo 7437
This theorem is referenced by:  tposconst  8290  mat0op  22426  matsc  22457  mdetrsca2  22611  smadiadetglem2  22679  fedgmullem2  33682
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