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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 3501. 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 38513. See
also the comment of df-rels 38486.
Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 38514, it keeps restriction of I: this is why the very similar definitions df-refs 38511, df-syms 38543 and df-trs 38573 diverge when we switch from (general) sets to relations in dfrefrels2 38514, dfsymrels2 38546 and dftrrels2 38576. (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 38512 | . 2 ⊢ RefRels = ( Refs ∩ Rels ) | |
| 2 | df-refs 38511 | . 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 3485 | . . . 4 ⊢ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V |
| 5 | brssr 38502 | . . . 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 38491 | . . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)) | |
| 8 | 7 | elv 3485 | . . . . 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 4016 | . . 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 38254 | 1 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟} |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 I cid 5577 × cxp 5683 dom cdm 5685 ran crn 5686 Rels crels 38184 S cssr 38185 Refs crefs 38186 RefRels crefrels 38187 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-rels 38486 df-ssr 38499 df-refs 38511 df-refrels 38512 |
| This theorem is referenced by: dfrefrels3 38515 elrefrels2 38519 refsymrels2 38566 refrelsredund4 38633 |
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