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Theorem fconstmpo 5832
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 4554 . 2 ((𝐴 × 𝐵) × {𝐶}) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶)
2 eqidd 2116 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐶)
32mpompt 5829 . 2 (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐶)
41, 3eqtri 2136 1 ((𝐴 × 𝐵) × {𝐶}) = (𝑥𝐴, 𝑦𝐵𝐶)
Colors of variables: wff set class
Syntax hints:   = wceq 1314  {csn 3495  cop 3498  cmpt 3957   × cxp 4505  cmpo 5742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-iun 3783  df-opab 3958  df-mpt 3959  df-xp 4513  df-rel 4514  df-oprab 5744  df-mpo 5745
This theorem is referenced by: (None)
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