Theorem dfcnvrefrels2 39343

Description: Alternate definition of the class of converse reflexive relations. See the comment of dfrefrels2 39328. (Contributed by Peter Mazsa, 21-Jul-2021.)

Assertion

Ref	Expression
dfcnvrefrels2	⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}

Proof of Theorem dfcnvrefrels2

Step	Hyp	Ref	Expression
1		df-cnvrefrels 39341	. 2 ⊢ CnvRefRels = ( CnvRefs ∩ Rels )
2		df-cnvrefs 39340	. 2 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3		dmexg 7901	. . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V)
4	3	elv 3458	. . . . 5 ⊢ dom 𝑟 ∈ V
5		rnexg 7902	. . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V)
6	5	elv 3458	. . . . 5 ⊢ ran 𝑟 ∈ V
7	4, 6	xpex 7755	. . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V
8		inex2g 5287	. . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
9		brcnvssr 39321	. . . 4 ⊢ (( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V → (( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))))
10	7, 8, 9	mp2b 10	. . 3 ⊢ (( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)))
11		elrels6 39180	. . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
12	11	elv 3458	. . . . 5 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
13	12	biimpi 219	. . . 4 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
14	13	sseq1d 3965	. . 3 ⊢ (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))))
15	10, 14	bitrid 286	. 2 ⊢ (𝑟 ∈ Rels → (( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))))
16	1, 2, 15	abeqinbi 38990	1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902   class class class wbr 5107   I cid 5553   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   Rels crels 38920   S cssr 38921   CnvRefs ccnvrefs 38925   CnvRefRels ccnvrefrels 38926

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 2734  ax-sep 5255  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-rels 39175  df-ssr 39313  df-cnvrefs 39340  df-cnvrefrels 39341

This theorem is used by:  elcnvrefrels2  39349