Theorem dfrefrels2 38469

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 7951) 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 3509. 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 38468. See also the comment of df-rels 38441.

Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 38469, it keeps restriction of I: this is why the very similar definitions df-refs 38466, df-syms 38498 and df-trs 38528 diverge when we switch from (general) sets to relations in dfrefrels2 38469, dfsymrels2 38501 and dftrrels2 38531. (Contributed by Peter Mazsa, 20-Jul-2019.)

Assertion

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

Proof of Theorem dfrefrels2

Step	Hyp	Ref	Expression
1		df-refrels 38467	. 2 ⊢ RefRels = ( Refs ∩ Rels )
2		df-refs 38466	. 2 ⊢ Refs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3		inex1g 5337	. . . . 5 ⊢ (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
4	3	elv 3493	. . . 4 ⊢ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V
5		brssr 38457	. . . 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 38446	. . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
8	7	elv 3493	. . . . 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 4041	. . 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 38209	1 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}

Syntax hints:   ↔ wb 206   = wceq 1537   ∈ wcel 2108  {crab 3443  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976   class class class wbr 5166   I cid 5592   × cxp 5698  dom cdm 5700  ran crn 5701   Rels crels 38137   S cssr 38138   Refs crefs 38139   RefRels crefrels 38140

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-rels 38441  df-ssr 38454  df-refs 38466  df-refrels 38467

This theorem is referenced by:  dfrefrels3  38470  elrefrels2  38474  refsymrels2  38521  refrelsredund4  38588