Theorem dfcnvrefrels3 38812

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 38809	. . 3 ⊢ CnvRefRels = ( CnvRefs ∩ Rels )
2		df-cnvrefs 38808	. . 3 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3	1, 2	abeqin 38457	. 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
4		dmexg 7845	. . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V)
5	4	elv 3446	. . . . 5 ⊢ dom 𝑟 ∈ V
6		rnexg 7846	. . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V)
7	6	elv 3446	. . . . 5 ⊢ ran 𝑟 ∈ V
8	5, 7	xpex 7700	. . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V
9		inex2g 5266	. . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
10		brcnvssr 38789	. . . 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 38525	. . 3 ⊢ ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦))
13	11, 12	bitri 275	. 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 3408	1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)}

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3400  Vcvv 3441   ∩ cin 3901   ⊆ wss 3902   class class class wbr 5099   I cid 5519   × cxp 5623  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   Rels crels 38388   S cssr 38389   CnvRefs ccnvrefs 38393   CnvRefRels ccnvrefrels 38394

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 2709  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-ssr 38781  df-cnvrefs 38808  df-cnvrefrels 38809

This theorem is referenced by:  elcnvrefrels3  38818