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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfrefrels2 | Structured version Visualization version GIF version |
Description: Alternate definition of
the class of reflexive relations. This is a 0-ary
class constant, which is recommended for definitions (see the 1.
Guideline at https://us.metamath.org/ileuni/mathbox.html).
Proper
classes (like I, see iprc 7933)
are not elements of this (or any)
class: if a class is an element of another class, it is not a proper class
but a set, see elex 3498. So if we use 0-ary constant classes as our
main
definitions, they are valid only for sets, not for proper classes. For
proper classes we use predicate-type definitions like df-refrel 38493. See
also the comment of df-rels 38466.
Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 38494, it keeps restriction of I: this is why the very similar definitions df-refs 38491, df-syms 38523 and df-trs 38553 diverge when we switch from (general) sets to relations in dfrefrels2 38494, dfsymrels2 38526 and dftrrels2 38556. (Contributed by Peter Mazsa, 20-Jul-2019.) |
Ref | Expression |
---|---|
dfrefrels2 | ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-refrels 38492 | . 2 ⊢ RefRels = ( Refs ∩ Rels ) | |
2 | df-refs 38491 | . 2 ⊢ Refs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} | |
3 | inex1g 5324 | . . . . 5 ⊢ (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V) | |
4 | 3 | elv 3482 | . . . 4 ⊢ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V |
5 | brssr 38482 | . . . 4 ⊢ ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V → (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)))) | |
6 | 4, 5 | ax-mp 5 | . . 3 ⊢ (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟))) |
7 | elrels6 38471 | . . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)) | |
8 | 7 | elv 3482 | . . . . 5 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
9 | 8 | biimpi 216 | . . . 4 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
10 | 9 | sseq2d 4027 | . . 3 ⊢ (𝑟 ∈ Rels → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟)) |
11 | 6, 10 | bitrid 283 | . 2 ⊢ (𝑟 ∈ Rels → (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟)) |
12 | 1, 2, 11 | abeqinbi 38234 | 1 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1536 ∈ wcel 2105 {crab 3432 Vcvv 3477 ∩ cin 3961 ⊆ wss 3962 class class class wbr 5147 I cid 5581 × cxp 5686 dom cdm 5688 ran crn 5689 Rels crels 38163 S cssr 38164 Refs crefs 38165 RefRels crefrels 38166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-rels 38466 df-ssr 38479 df-refs 38491 df-refrels 38492 |
This theorem is referenced by: dfrefrels3 38495 elrefrels2 38499 refsymrels2 38546 refrelsredund4 38613 |
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