Theorem dfcnvrefrels2 38929

Description: Alternate definition of the class of converse reflexive relations. See the comment of dfrefrels2 38914. (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 38927	. 2 ⊢ CnvRefRels = ( CnvRefs ∩ Rels )
2		df-cnvrefs 38926	. 2 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3		dmexg 7852	. . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V)
4	3	elv 3434	. . . . 5 ⊢ dom 𝑟 ∈ V
5		rnexg 7853	. . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V)
6	5	elv 3434	. . . . 5 ⊢ ran 𝑟 ∈ V
7	4, 6	xpex 7707	. . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V
8		inex2g 5261	. . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
9		brcnvssr 38907	. . . 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 38766	. . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
12	11	elv 3434	. . . . 5 ⊢ (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
13	12	biimpi 216	. . . 4 ⊢ (𝑟 ∈ Rels → (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟)
14	13	sseq1d 3953	. . 3 ⊢ (𝑟 ∈ Rels → ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))))
15	10, 14	bitrid 283	. 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 38576	1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {crab 3389  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889   class class class wbr 5085   I cid 5525   × cxp 5629  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   Rels crels 38506   S cssr 38507   CnvRefs ccnvrefs 38511   CnvRefRels ccnvrefrels 38512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-rels 38761  df-ssr 38899  df-cnvrefs 38926  df-cnvrefrels 38927

This theorem is referenced by:  elcnvrefrels2  38935