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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 5226 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
| 2 | 1 | relopabiv 5830 | 1 ⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 ↦ cmpt 5225 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-opab 5206 df-mpt 5226 df-xp 5691 df-rel 5692 |
| This theorem is referenced by: fmptco 7149 swrd0 14696 pmtrsn 19537 00lsp 20979 fmptcof2 32667 dfbigcup2 35900 imageval 35931 iscard4 43546 |
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