Theorem dfrefrels2 39493

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 7912) 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 3472. 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 39492. See also the comment of df-rels 39340.

Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 39493, it keeps restriction of I: this is why the very similar definitions df-refs 39490, df-syms 39522 and df-trs 39556 diverge when we switch from (general) sets to relations in dfrefrels2 39493, dfsymrels2 39525 and dftrrels2 39559. (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 39491	. 2 ⊢ RefRels = ( Refs ∩ Rels )
2		df-refs 39490	. 2 ⊢ Refs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3		inex1g 5279	. . . . 5 ⊢ (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
4	3	elv 3456	. . . 4 ⊢ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V
5		brssr 39481	. . . 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 39345	. . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
8	7	elv 3456	. . . . 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 3963	. . 3 ⊢ (𝑟 ∈ Rels → (( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟))
11	6, 10	bitrid 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 39155	1 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2145  {crab 3413  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899   class class class wbr 5103   I cid 5545   × cxp 5649  dom cdm 5651  ran crn 5652   Rels crels 39085   S cssr 39086   Refs crefs 39087   RefRels crefrels 39088

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-rels 39340  df-ssr 39478  df-refs 39490  df-refrels 39491

This theorem is used by:  dfrefrels3  39494  elrefrels2  39498  refsymrels2  39549  refrelsredund4  39616