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Theorem fconstmpo 7550
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 5751 . 2 ((𝐴 × 𝐵) × {𝐶}) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶)
2 eqidd 2736 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐶)
32mpompt 7547 . 2 (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐶)
41, 3eqtri 2763 1 ((𝐴 × 𝐵) × {𝐶}) = (𝑥𝐴, 𝑦𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  {csn 4631  cop 4637  cmpt 5231   × cxp 5687  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-iun 4998  df-opab 5211  df-mpt 5232  df-xp 5695  df-rel 5696  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  tposconst  8288  mat0op  22441  matsc  22472  mdetrsca2  22626  smadiadetglem2  22694  fedgmullem2  33658
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