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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 5139 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
2 | 1 | relopabi 5688 | 1 ⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∈ wcel 2110 ↦ cmpt 5138 Rel wrel 5554 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5121 df-mpt 5139 df-xp 5555 df-rel 5556 |
This theorem is referenced by: fmptco 6885 swrd0 14014 pmtrsn 18641 00lsp 19747 fmptcof2 30396 dfbigcup2 33355 imageval 33386 iscard4 39893 |
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