Theorem dfcnvrefrels2 34899

Description: Alternate definition of the class of converse reflexive relations. Cf. the comment of dfrefrels2 34886. (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 34897	. 2 ⊢ CnvRefRels = ( CnvRefs ∩ Rels )
2		df-cnvrefs 34896	. 2 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3		dmexg 7375	. . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V)
4	3	elv 3401	. . . . 5 ⊢ dom 𝑟 ∈ V
5		rnexg 7376	. . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V)
6	5	elv 3401	. . . . 5 ⊢ ran 𝑟 ∈ V
7	4, 6	xpex 7240	. . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V
8		inex2ALTV 34728	. . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
9		brcnvssr 34879	. . . 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 34863	. . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
12	11	elv 3401	. . . . 5 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
13	12	biimpi 208	. . . 4 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
14	13	sseq1d 3850	. . 3 ⊢ (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))))
15	10, 14	syl5bb 275	. 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 34648	1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}

Syntax hints:   ↔ wb 198   = wceq 1601   ∈ wcel 2106  {crab 3093  Vcvv 3397   ∩ cin 3790   ⊆ wss 3791   class class class wbr 4886   I cid 5260   × cxp 5353  ^◡ccnv 5354  dom cdm 5355  ran crn 5356   Rels crels 34603   S cssr 34604   CnvRefs ccnvrefs 34608   CnvRefRels ccnvrefrels 34609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-xp 5361  df-rel 5362  df-cnv 5363  df-dm 5365  df-rn 5366  df-res 5367  df-rels 34858  df-ssr 34871  df-cnvrefs 34896  df-cnvrefrels 34897

This theorem is referenced by:  elcnvrefrels2  34903