Theorem fconstmpo 7480

Description: Representation of a constant operation using the mapping operation. (Contributed by SO, 11-Jul-2018.)

Assertion

Ref	Expression
fconstmpo	⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

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.

Step	Hyp	Ref	Expression
1		fconstmpt 5687	. 2 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶)
2		eqidd 2741	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐶)
3	2	mpompt 7477	. 2 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	1, 3	eqtri 2763	1 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   = wceq 1547  {csn 4562  ⟨cop 4568   ↦ cmpt 5160   × cxp 5623   ∈ cmpo 7365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-iun 4930  df-opab 5142  df-mpt 5161  df-xp 5631  df-rel 5632  df-oprab 7367  df-mpo 7368

This theorem is referenced by:  tposconst  8211  mat0op  22409  matsc  22440  mdetrsca2  22594  smadiadetglem2  22662  fedgmullem2  33821