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Theorem dfrefrels2 35906
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 7604) 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 3462. 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 35905. See also the comment of df-rels 35878.

Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 35906, it keeps restriction of I: this is why the very similar definitions df-refs 35903, df-syms 35931 and df-trs 35961 diverge when we switch from (general) sets to relations in dfrefrels2 35906, dfsymrels2 35934 and dftrrels2 35964. (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 35904 . 2 RefRels = ( Refs ∩ Rels )
2 df-refs 35903 . 2 Refs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3 inex1g 5190 . . . . 5 (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
43elv 3449 . . . 4 (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V
5 brssr 35894 . . . 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 35883 . . . . . 6 (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
87elv 3449 . . . . 5 (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
98biimpi 219 . . . 4 (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
109sseq2d 3950 . . 3 (𝑟 ∈ Rels → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟))
116, 10syl5bb 286 . 2 (𝑟 ∈ Rels → (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟))
121, 2, 11abeqinbi 35668 1 RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wcel 2112  {crab 3113  Vcvv 3444  cin 3883  wss 3884   class class class wbr 5033   I cid 5427   × cxp 5521  dom cdm 5523  ran crn 5524   Rels crels 35608   S cssr 35609   Refs crefs 35610   RefRels crefrels 35611
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-rels 35878  df-ssr 35891  df-refs 35903  df-refrels 35904
This theorem is referenced by:  dfrefrels3  35907  elrefrels2  35910  refsymrels2  35954  refrelsredund4  36020
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