Theorem dfcnvrefrels3 39461

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 39458	. . 3 ⊢ CnvRefRels = ( CnvRefs ∩ Rels )
2		df-cnvrefs 39457	. . 3 ⊢ CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
3	1, 2	abeqin 39106	. 2 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟))◡ S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
4		dmexg 7896	. . . . . 6 ⊢ (𝑟 ∈ V → dom 𝑟 ∈ V)
5	4	elv 3455	. . . . 5 ⊢ dom 𝑟 ∈ V
6		rnexg 7897	. . . . . 6 ⊢ (𝑟 ∈ V → ran 𝑟 ∈ V)
7	6	elv 3455	. . . . 5 ⊢ ran 𝑟 ∈ V
8	5, 7	xpex 7750	. . . 4 ⊢ (dom 𝑟 × ran 𝑟) ∈ V
9		inex2g 5279	. . . 4 ⊢ ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
10		brcnvssr 39438	. . . 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 39174	. . 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 3420	1 ⊢ CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟∀𝑦 ∈ ran 𝑟(𝑥𝑟𝑦 → 𝑥 = 𝑦)}

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412  Vcvv 3450   ∩ cin 3897   ⊆ wss 3898   class class class wbr 5102   I cid 5541   × cxp 5645  ^◡ccnv 5646  dom cdm 5647  ran crn 5648   Rels crels 39037   S cssr 39038   CnvRefs ccnvrefs 39042   CnvRefRels ccnvrefrels 39043

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pow 5326  ax-pr 5390  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658  df-ssr 39430  df-cnvrefs 39457  df-cnvrefrels 39458

This theorem is used by:  elcnvrefrels3  39467