Theorem dfcnvrefrels2 38777

Description: Alternate definition of the class of converse reflexive relations. See the comment of dfrefrels2 38762. (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 38775	. 2 ⊢ CnvRefRels = ( CnvRefs ∩ Rels )
2		df-cnvrefs 38774	. 2 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3		dmexg 7843	. . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V)
4	3	elv 3445	. . . . 5 ⊢ dom 𝑟 ∈ V
5		rnexg 7844	. . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V)
6	5	elv 3445	. . . . 5 ⊢ ran 𝑟 ∈ V
7	4, 6	xpex 7698	. . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V
8		inex2g 5265	. . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
9		brcnvssr 38755	. . . 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 38626	. . . . . 6 ⊢ (𝑟 ∈ V → (𝑟 ∈ Rels ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) = 𝑟))
12	11	elv 3445	. . . . 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 3965	. . 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 38447	1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ 𝑟 ⊆ ( I ∩ (dom 𝑟 × ran 𝑟))}

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2113  {crab 3399  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901   class class class wbr 5098   I cid 5518   × cxp 5622  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   Rels crels 38381   S cssr 38382   CnvRefs ccnvrefs 38386   CnvRefRels ccnvrefrels 38387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-rels 38621  df-ssr 38747  df-cnvrefs 38774  df-cnvrefrels 38775

This theorem is referenced by:  elcnvrefrels2  38783