Theorem fconstmpo 7517

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 5713	. 2 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶)
2		eqidd 2766	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐶)
3	2	mpompt 7514	. 2 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
4	1, 3	eqtri 2788	1 ⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   = wceq 1563  {csn 4585  ⟨cop 4591   ↦ cmpt 5185   × cxp 5649   ∈ cmpo 7402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-iun 4953  df-opab 5167  df-mpt 5186  df-xp 5657  df-rel 5658  df-oprab 7404  df-mpo 7405

This theorem is referenced by:  tposconst  8248  mat0op  22533  matsc  22564  mdetrsca2  22718  smadiadetglem2  22786  fedgmullem2  33932