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Theorem fconstmpo 7336
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 5620 . 2 ((𝐴 × 𝐵) × {𝐶}) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶)
2 eqidd 2739 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐶)
32mpompt 7333 . 2 (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐶)
41, 3eqtri 2766 1 ((𝐴 × 𝐵) × {𝐶}) = (𝑥𝐴, 𝑦𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  {csn 4550  cop 4556  cmpt 5144   × cxp 5558  cmpo 7224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5201  ax-nul 5208  ax-pr 5331
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3417  df-sbc 3704  df-csb 3821  df-dif 3878  df-un 3880  df-in 3882  df-ss 3892  df-nul 4247  df-if 4449  df-sn 4551  df-pr 4553  df-op 4557  df-iun 4915  df-opab 5125  df-mpt 5145  df-xp 5566  df-rel 5567  df-oprab 7226  df-mpo 7227
This theorem is referenced by:  tposconst  8015  mat0op  21329  matsc  21360  mdetrsca2  21514  smadiadetglem2  21582  fedgmullem2  31438
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