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Theorem fconstmpt2 7032
 Description: Representation of a constant operation using the mapping operation. (Contributed by SO, 11-Jul-2018.)
Assertion
Ref Expression
fconstmpt2 ((𝐴 × 𝐵) × {𝐶}) = (𝑥𝐴, 𝑦𝐵𝐶)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem fconstmpt2
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 fconstmpt 5411 . 2 ((𝐴 × 𝐵) × {𝐶}) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶)
2 eqidd 2779 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐶)
32mpt2mpt 7029 . 2 (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐶)
41, 3eqtri 2802 1 ((𝐴 × 𝐵) × {𝐶}) = (𝑥𝐴, 𝑦𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1601  {csn 4398  ⟨cop 4404   ↦ cmpt 4965   × cxp 5353   ↦ cmpt2 6924 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-iun 4755  df-opab 4949  df-mpt 4966  df-xp 5361  df-rel 5362  df-oprab 6926  df-mpt2 6927 This theorem is referenced by:  tposconst  7672  mat0op  20629  matsc  20661  mdetrsca2  20815  smadiadetglem2  20884
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