Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfdisjALTV Structured version   Visualization version   GIF version

Theorem dfdisjALTV 37225
Description: Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs 37216 why we need disjoint relations instead of converse functions anyway. (Contributed by Peter Mazsa, 27-Jul-2021.)
Assertion
Ref Expression
dfdisjALTV ( Disj 𝑅 ↔ ( FunALTV 𝑅 ∧ Rel 𝑅))

Proof of Theorem dfdisjALTV
StepHypRef Expression
1 df-disjALTV 37217 . 2 ( Disj 𝑅 ↔ ( CnvRefRel ≀ 𝑅 ∧ Rel 𝑅))
2 relcnv 6060 . . . 4 Rel 𝑅
3 df-funALTV 37194 . . . 4 ( FunALTV 𝑅 ↔ ( CnvRefRel ≀ 𝑅 ∧ Rel 𝑅))
42, 3mpbiran2 709 . . 3 ( FunALTV 𝑅 ↔ CnvRefRel ≀ 𝑅)
54anbi1i 625 . 2 (( FunALTV 𝑅 ∧ Rel 𝑅) ↔ ( CnvRefRel ≀ 𝑅 ∧ Rel 𝑅))
61, 5bitr4i 278 1 ( Disj 𝑅 ↔ ( FunALTV 𝑅 ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  ccnv 5636  Rel wrel 5642  ccoss 36684   CnvRefRel wcnvrefrel 36693   FunALTV wfunALTV 36715   Disj wdisjALTV 36718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-in 3921  df-ss 3931  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-funALTV 37194  df-disjALTV 37217
This theorem is referenced by:  disjss  37243
  Copyright terms: Public domain W3C validator