Theorem mptrel 5765

Description: The maps-to notation always describes a binary relation. (Contributed by Scott Fenton, 16-Apr-2012.)

Assertion

Ref	Expression
mptrel	⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵)

Proof of Theorem mptrel

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt 5173	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
2	1	relopabiv 5760	1 ⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ↦ cmpt 5172  Rel wrel 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-ss 3919  df-opab 5154  df-mpt 5173  df-xp 5622  df-rel 5623

This theorem is referenced by:  fmptco  7062  swrd0  14563  pmtrsn  19429  00lsp  20912  fmptcof2  32634  dfbigcup2  35932  imageval  35963  iscard4  43565