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 5195	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
2	1	relopabiv 5809	1 ⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ↦ cmpt 5194  Rel wrel 5668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-ss 3923  df-opab 5176  df-mpt 5195  df-xp 5669  df-rel 5670

This theorem is used by:  fmptco  7129  swrd0  14714  pmtrsn  19613  00lsp  21132  fmptcof2  33049  dfbigcup2  36402  imageval  36433  relqmap  39134  iscard4  44292