Theorem mptrel 5814

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 5194	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
2	1	relopabiv 5809	1 ⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ↦ cmpt 5193  Rel wrel 5668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-ss 3923  df-opab 5175  df-mpt 5194  df-xp 5669  df-rel 5670

This theorem is referenced by:  fmptco  7127  swrd0  14698  pmtrsn  19590  00lsp  21083  fmptcof2  32983  dfbigcup2  36370  imageval  36401  relqmap  39082  iscard4  44242