Theorem mptrel 5775

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 5161	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
2	1	relopabiv 5770	1 ⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ↦ cmpt 5160  Rel wrel 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-ss 3907  df-opab 5142  df-mpt 5161  df-xp 5631  df-rel 5632

This theorem is referenced by:  fmptco  7078  swrd0  14619  pmtrsn  19492  00lsp  20978  fmptcof2  32756  dfbigcup2  36132  imageval  36163  relqmap  38826  iscard4  43984