Theorem mptrel 5823

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 5229	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
2	1	relopabiv 5818	1 ⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   ∧ wa 394   = wceq 1534   ∈ wcel 2099   ↦ cmpt 5228  Rel wrel 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-ss 3963  df-opab 5208  df-mpt 5229  df-xp 5680  df-rel 5681

This theorem is referenced by:  fmptco  7135  swrd0  14661  pmtrsn  19513  00lsp  20954  fmptcof2  32574  dfbigcup2  35736  imageval  35767  iscard4  43237