Theorem dfcnvrefrels3 36572

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 36569	. . 3 ⊢ CnvRefRels = ( CnvRefs ∩ Rels )
2		df-cnvrefs 36568	. . 3 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3	1, 2	abeqin 36319	. 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
4		dmexg 7724	. . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V)
5	4	elv 3428	. . . . 5 ⊢ dom 𝑟 ∈ V
6		rnexg 7725	. . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V)
7	6	elv 3428	. . . . 5 ⊢ ran 𝑟 ∈ V
8	5, 7	xpex 7581	. . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V
9		inex2g 5239	. . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
10		brcnvssr 36551	. . . 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 36378	. . 3 ⊢ ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦))
13	11, 12	bitri 274	. 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 36317	1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)}

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883   class class class wbr 5070   I cid 5479   × cxp 5578  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   Rels crels 36262   S cssr 36263   CnvRefs ccnvrefs 36267   CnvRefRels ccnvrefrels 36268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-ssr 36543  df-cnvrefs 36568  df-cnvrefrels 36569

This theorem is referenced by:  elcnvrefrels3  36576