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Theorem dfdisjALTV 37886
Description: Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs 37877 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 37878 . 2 ( Disj 𝑅 ↔ ( CnvRefRel ≀ 𝑅 ∧ Rel 𝑅))
2 relcnv 6102 . . . 4 Rel 𝑅
3 df-funALTV 37855 . . . 4 ( FunALTV 𝑅 ↔ ( CnvRefRel ≀ 𝑅 ∧ Rel 𝑅))
42, 3mpbiran2 706 . . 3 ( FunALTV 𝑅 ↔ CnvRefRel ≀ 𝑅)
54anbi1i 622 . 2 (( FunALTV 𝑅 ∧ Rel 𝑅) ↔ ( CnvRefRel ≀ 𝑅 ∧ Rel 𝑅))
61, 5bitr4i 277 1 ( Disj 𝑅 ↔ ( FunALTV 𝑅 ∧ Rel 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394  ccnv 5674  Rel wrel 5680  ccoss 37346   CnvRefRel wcnvrefrel 37355   FunALTV wfunALTV 37377   Disj wdisjALTV 37380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-in 3954  df-ss 3964  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-funALTV 37855  df-disjALTV 37878
This theorem is referenced by:  disjss  37904
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