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Theorem dfcnvrefrel3 38055
Description: Alternate definition of the converse reflexive relation predicate. A relation is converse reflexive iff: for all elements on its domain and range, if for an element of its domain and for an element of its range there is the relation between them, then the two elements are the same, cf. the comment of dfrefrel3 38040. (Contributed by Peter Mazsa, 25-Jul-2021.)
Assertion
Ref Expression
dfcnvrefrel3 ( CnvRefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥𝑅𝑦𝑥 = 𝑦) ∧ Rel 𝑅))
Distinct variable group:   𝑥,𝑅,𝑦

Proof of Theorem dfcnvrefrel3
StepHypRef Expression
1 df-cnvrefrel 38051 . 2 ( CnvRefRel 𝑅 ↔ ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅))
2 inxpssidinxp 37840 . . 3 ((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ↔ ∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥𝑅𝑦𝑥 = 𝑦))
32anbi1i 622 . 2 (((𝑅 ∩ (dom 𝑅 × ran 𝑅)) ⊆ ( I ∩ (dom 𝑅 × ran 𝑅)) ∧ Rel 𝑅) ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥𝑅𝑦𝑥 = 𝑦) ∧ Rel 𝑅))
41, 3bitri 274 1 ( CnvRefRel 𝑅 ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ ran 𝑅(𝑥𝑅𝑦𝑥 = 𝑦) ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wral 3051  cin 3940  wss 3941   class class class wbr 5144   I cid 5570   × cxp 5671  dom cdm 5673  ran crn 5674  Rel wrel 5678   CnvRefRel wcnvrefrel 37710
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-pr 5424
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-sn 4626  df-pr 4628  df-op 4632  df-br 5145  df-opab 5207  df-id 5571  df-xp 5679  df-rel 5680  df-cnvrefrel 38051
This theorem is referenced by: (None)
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