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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 7680)
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 3419. 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 36324. See
also the comment of df-rels 36297.
Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 36325, it keeps restriction of I: this is why the very similar definitions df-refs 36322, df-syms 36350 and df-trs 36380 diverge when we switch from (general) sets to relations in dfrefrels2 36325, dfsymrels2 36353 and dftrrels2 36383. (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 36323 | . 2 ⊢ RefRels = ( Refs ∩ Rels ) | |
2 | df-refs 36322 | . 2 ⊢ Refs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))} | |
3 | inex1g 5201 | . . . . 5 ⊢ (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V) | |
4 | 3 | elv 3407 | . . . 4 ⊢ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V |
5 | brssr 36313 | . . . 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 36302 | . . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)) | |
8 | 7 | elv 3407 | . . . . 5 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
9 | 8 | biimpi 219 | . . . 4 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟) |
10 | 9 | sseq2d 3923 | . . 3 ⊢ (𝑟 ∈ Rels → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟)) |
11 | 6, 10 | syl5bb 286 | . 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 36087 | 1 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2110 {crab 3058 Vcvv 3401 ∩ cin 3856 ⊆ wss 3857 class class class wbr 5043 I cid 5443 × cxp 5538 dom cdm 5540 ran crn 5541 Rels crels 36029 S cssr 36030 Refs crefs 36031 RefRels crefrels 36032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-br 5044 df-opab 5106 df-xp 5546 df-rel 5547 df-cnv 5548 df-dm 5550 df-rn 5551 df-res 5552 df-rels 36297 df-ssr 36310 df-refs 36322 df-refrels 36323 |
This theorem is referenced by: dfrefrels3 36326 elrefrels2 36329 refsymrels2 36373 refrelsredund4 36439 |
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