Theorem dfrefrels2 37383

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 7904) 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 3493. 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 37382. See also the comment of df-rels 37355.

Note that while elementhood in the class of relations cancels restriction of 𝑟 in dfrefrels2 37383, it keeps restriction of I: this is why the very similar definitions df-refs 37380, df-syms 37412 and df-trs 37442 diverge when we switch from (general) sets to relations in dfrefrels2 37383, dfsymrels2 37415 and dftrrels2 37445. (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 37381	. 2 ⊢ RefRels = ( Refs ∩ Rels )
2		df-refs 37380	. 2 ⊢ Refs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3		inex1g 5320	. . . . 5 ⊢ (𝑟 ∈ V → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
4	3	elv 3481	. . . 4 ⊢ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ∈ V
5		brssr 37371	. . . 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 37360	. . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
8	7	elv 3481	. . . . 5 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
9	8	biimpi 215	. . . 4 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
10	9	sseq2d 4015	. . 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 37121	1 ⊢ RefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) ⊆ 𝑟}

Syntax hints:   ↔ wb 205   = wceq 1542   ∈ wcel 2107  {crab 3433  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949   class class class wbr 5149   I cid 5574   × cxp 5675  dom cdm 5677  ran crn 5678   Rels crels 37045   S cssr 37046   Refs crefs 37047   RefRels crefrels 37048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-rels 37355  df-ssr 37368  df-refs 37380  df-refrels 37381

This theorem is referenced by:  dfrefrels3  37384  elrefrels2  37388  refsymrels2  37435  refrelsredund4  37502