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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 7903)
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 3492. 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 37377. See
also the comment of df-rels 37350.
Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 37378, it keeps restriction of I: this is why the very similar definitions df-refs 37375, df-syms 37407 and df-trs 37437 diverge when we switch from (general) sets to relations in dfrefrels2 37378, dfsymrels2 37410 and dftrrels2 37440. (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 37376 | . 2 ⊢ RefRels = ( Refs ∩ Rels ) | |
2 | df-refs 37375 | . 2 ⊢ Refs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} | |
3 | inex1g 5319 | . . . . 5 ⊢ (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V) | |
4 | 3 | elv 3480 | . . . 4 ⊢ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V |
5 | brssr 37366 | . . . 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 37355 | . . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)) | |
8 | 7 | elv 3480 | . . . . 5 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
9 | 8 | biimpi 215 | . . . 4 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
10 | 9 | sseq2d 4014 | . . 3 ⊢ (𝑟 ∈ Rels → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟)) |
11 | 6, 10 | bitrid 282 | . 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 37116 | 1 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 {crab 3432 Vcvv 3474 ∩ cin 3947 ⊆ wss 3948 class class class wbr 5148 I cid 5573 × cxp 5674 dom cdm 5676 ran crn 5677 Rels crels 37040 S cssr 37041 Refs crefs 37042 RefRels crefrels 37043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-rels 37350 df-ssr 37363 df-refs 37375 df-refrels 37376 |
This theorem is referenced by: dfrefrels3 37379 elrefrels2 37383 refsymrels2 37430 refrelsredund4 37497 |
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