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Theorem dfcnvrefrels3 38053
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 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦)}
Distinct variable group:   𝑥,𝑟,𝑦

Proof of Theorem dfcnvrefrels3
StepHypRef Expression
1 df-cnvrefrels 38050 . . 3 CnvRefRels = ( CnvRefs ∩ Rels )
2 df-cnvrefs 38049 . . 3 CnvRefs = {𝑟 ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
31, 2abeqin 37776 . 2 CnvRefRels = {𝑟 ∈ Rels ∣ ( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟))}
4 dmexg 7903 . . . . . 6 (𝑟 ∈ V → dom 𝑟 ∈ V)
54elv 3469 . . . . 5 dom 𝑟 ∈ V
6 rnexg 7904 . . . . . 6 (𝑟 ∈ V → ran 𝑟 ∈ V)
76elv 3469 . . . . 5 ran 𝑟 ∈ V
85, 7xpex 7750 . . . 4 (dom 𝑟 × ran 𝑟) ∈ V
9 inex2g 5316 . . . 4 ((dom 𝑟 × ran 𝑟) ∈ V → ( I ∩ (dom 𝑟 × ran 𝑟)) ∈ V)
10 brcnvssr 38030 . . . 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))))
118, 9, 10mp2b 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 37840 . . 3 ((𝑟 ∩ (dom 𝑟 × ran 𝑟)) ⊆ ( I ∩ (dom 𝑟 × ran 𝑟)) ↔ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦))
1311, 12bitri 274 . 2 (( I ∩ (dom 𝑟 × ran 𝑟)) S (𝑟 ∩ (dom 𝑟 × ran 𝑟)) ↔ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦))
143, 13rabbieq 3428 1 CnvRefRels = {𝑟 ∈ Rels ∣ ∀𝑥 ∈ dom 𝑟𝑦 ∈ ran 𝑟(𝑥𝑟𝑦𝑥 = 𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  wral 3051  {crab 3419  Vcvv 3463  cin 3940  wss 3941   class class class wbr 5144   I cid 5570   × cxp 5671  ccnv 5672  dom cdm 5673  ran crn 5674   Rels crels 37703   S cssr 37704   CnvRefs ccnvrefs 37708   CnvRefRels ccnvrefrels 37709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-dm 5683  df-rn 5684  df-ssr 38022  df-cnvrefs 38049  df-cnvrefrels 38050
This theorem is referenced by:  elcnvrefrels3  38059
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