Theorem dfdisjALTV 38714

Description: Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs 38705 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

Step	Hyp	Ref	Expression
1		df-disjALTV 38706	. 2 ⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))
2		relcnv 6122	. . . 4 ⊢ Rel ◡𝑅
3		df-funALTV 38683	. . . 4 ⊢ ( FunALTV ◡𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel ◡𝑅))
4	2, 3	mpbiran2 710	. . 3 ⊢ ( FunALTV ◡𝑅 ↔ CnvRefRel ≀ ◡𝑅)
5	4	anbi1i 624	. 2 ⊢ (( FunALTV ◡𝑅 ∧ Rel 𝑅) ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))
6	1, 5	bitr4i 278	1 ⊢ ( Disj 𝑅 ↔ ( FunALTV ◡𝑅 ∧ Rel 𝑅))

Syntax hints:   ↔ wb 206   ∧ wa 395  ^◡ccnv 5684  Rel wrel 5690   ≀ ccoss 38182   CnvRefRel wcnvrefrel 38191   FunALTV wfunALTV 38213   Disj wdisjALTV 38216

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-funALTV 38683  df-disjALTV 38706

This theorem is referenced by:  disjss  38732