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Theorem dfdisjALTV 37583
Description: Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs 37574 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 37575 . 2 ( Disj 𝑅 ↔ ( CnvRefRel ≀ 𝑅 ∧ Rel 𝑅))
2 relcnv 6104 . . . 4 Rel 𝑅
3 df-funALTV 37552 . . . 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 5676  Rel wrel 5682  ccoss 37043   CnvRefRel wcnvrefrel 37052   FunALTV wfunALTV 37074   Disj wdisjALTV 37077
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 3477  df-in 3956  df-ss 3966  df-opab 5212  df-xp 5683  df-rel 5684  df-cnv 5685  df-funALTV 37552  df-disjALTV 37575
This theorem is referenced by:  disjss  37601
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