Theorem mptrel 5802

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 5186	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
2	1	relopabiv 5797	1 ⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ↦ cmpt 5185  Rel wrel 5656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-ss 3924  df-opab 5167  df-mpt 5186  df-xp 5657  df-rel 5658

This theorem is referenced by:  fmptco  7115  swrd0  14684  pmtrsn  19577  00lsp  21068  fmptcof2  32910  dfbigcup2  36255  imageval  36286  relqmap  38958  iscard4  44116