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Mirrors > Home > MPE Home > Th. List > mptrel | Structured version Visualization version GIF version |
Description: The maps-to notation always describes a binary relation. (Contributed by Scott Fenton, 16-Apr-2012.) |
Ref | Expression |
---|---|
mptrel | ⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpt 5232 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
2 | 1 | relopabiv 5833 | 1 ⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 ↦ cmpt 5231 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-opab 5211 df-mpt 5232 df-xp 5695 df-rel 5696 |
This theorem is referenced by: fmptco 7149 swrd0 14693 pmtrsn 19552 00lsp 20997 fmptcof2 32674 dfbigcup2 35881 imageval 35912 iscard4 43523 |
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