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Theorem dfrefrels2 36788
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 7828) 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 3459. 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 36787. See also the comment of df-rels 36760.

Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 36788, it keeps restriction of I: this is why the very similar definitions df-refs 36785, df-syms 36817 and df-trs 36847 diverge when we switch from (general) sets to relations in dfrefrels2 36788, dfsymrels2 36820 and dftrrels2 36850. (Contributed by Peter Mazsa, 20-Jul-2019.)

Assertion
Ref Expression
dfrefrels2 RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}

Proof of Theorem dfrefrels2
StepHypRef Expression
1 df-refrels 36786 . 2 RefRels = ( Refs ∩ Rels )
2 df-refs 36785 . 2 Refs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3 inex1g 5263 . . . . 5 (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
43elv 3447 . . . 4 (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V
5 brssr 36776 . . . 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))))
64, 5ax-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 36765 . . . . . 6 (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
87elv 3447 . . . . 5 (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
98biimpi 215 . . . 4 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
109sseq2d 3964 . . 3 (𝑟 ∈ Rels → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟))
116, 10bitrid 282 . 2 (𝑟 ∈ Rels → (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟))
121, 2, 11abeqinbi 36526 1 RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1540  wcel 2105  {crab 3403  Vcvv 3441  cin 3897  wss 3898   class class class wbr 5092   I cid 5517   × cxp 5618  dom cdm 5620  ran crn 5621   Rels crels 36448   S cssr 36449   Refs crefs 36450   RefRels crefrels 36451
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-xp 5626  df-rel 5627  df-cnv 5628  df-dm 5630  df-rn 5631  df-res 5632  df-rels 36760  df-ssr 36773  df-refs 36785  df-refrels 36786
This theorem is referenced by:  dfrefrels3  36789  elrefrels2  36793  refsymrels2  36840  refrelsredund4  36907
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