Theorem dfcnvrefrels2 39112

Description: Alternate definition of the class of converse reflexive relations. See the comment of dfrefrels2 39097. (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 39110	. 2 ⊢ CnvRefRels = ( CnvRefs ∩ Rels )
2		df-cnvrefs 39109	. 2 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3		dmexg 7884	. . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V)
4	3	elv 3461	. . . . 5 ⊢ dom 𝑟 ∈ V
5		rnexg 7885	. . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V)
6	5	elv 3461	. . . . 5 ⊢ ran 𝑟 ∈ V
7	4, 6	xpex 7738	. . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V
8		inex2g 5278	. . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
9		brcnvssr 39090	. . . 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 38949	. . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
12	11	elv 3461	. . . . 5 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
13	12	biimpi 218	. . . 4 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
14	13	sseq1d 3969	. . 3 ⊢ (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))))
15	10, 14	bitrid 285	. 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 38759	1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}

Syntax hints:   ↔ wb 208   = wceq 1562   ∈ wcel 2144  {crab 3416  Vcvv 3456   ∩ cin 3905   ⊆ wss 3906   class class class wbr 5102   I cid 5543   × cxp 5647  ^◡ccnv 5648  dom cdm 5649  ran crn 5650   Rels crels 38689   S cssr 38690   CnvRefs ccnvrefs 38694   CnvRefRels ccnvrefrels 38695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5655  df-rel 5656  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661  df-rels 38944  df-ssr 39082  df-cnvrefs 39109  df-cnvrefrels 39110

This theorem is referenced by:  elcnvrefrels2  39118