Theorem dfcnvrefrels3 39208

Description: Alternate definition of the class of converse reflexive relations. (Contributed by Peter Mazsa, 22-Jul-2019.)

Assertion

Ref	Expression
dfcnvrefrels3	⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)}

Distinct variable group:   𝑥,𝑟,𝑦

Proof of Theorem dfcnvrefrels3

Step	Hyp	Ref	Expression
1		df-cnvrefrels 39205	. . 3 ⊢ CnvRefRels = ( CnvRefs ∩ Rels )
2		df-cnvrefs 39204	. . 3 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3	1, 2	abeqin 38853	. 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
4		dmexg 7901	. . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V)
5	4	elv 3467	. . . . 5 ⊢ dom 𝑟 ∈ V
6		rnexg 7902	. . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V)
7	6	elv 3467	. . . . 5 ⊢ ran 𝑟 ∈ V
8	5, 7	xpex 7755	. . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V
9		inex2g 5294	. . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
10		brcnvssr 39185	. . . 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))))
11	8, 9, 10	mp2b 10	. . 3 ⊢ (( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)))
12		inxpssidinxp 38921	. . 3 ⊢ ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦))
13	11, 12	bitri 278	. 2 ⊢ (( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦))
14	3, 13	rabbieq 3431	1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)}

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1568   ∈ wcel 2150  ∀wral 3086  {crab 3423  Vcvv 3462   ∩ cin 3912   ⊆ wss 3913   class class class wbr 5114   I cid 5559   × cxp 5663  ^◡ccnv 5664  dom cdm 5665  ran crn 5666   Rels crels 38784   S cssr 38785   CnvRefs ccnvrefs 38789   CnvRefRels ccnvrefrels 38790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-dm 5675  df-rn 5676  df-ssr 39177  df-cnvrefs 39204  df-cnvrefrels 39205

This theorem is referenced by:  elcnvrefrels3  39214