Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5147 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
2 | 1 | relopabi 5694 | 1 ⊢ Rel (𝑥 ∈ 𝐴 ↦ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5146 Rel wrel 5560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 df-mpt 5147 df-xp 5561 df-rel 5562 |
This theorem is referenced by: fmptco 6891 swrd0 14020 pmtrsn 18647 00lsp 19753 fmptcof2 30402 dfbigcup2 33360 imageval 33391 iscard4 39920 |
Copyright terms: Public domain | W3C validator |