Theorem dfdisjALTV 39298

Description: Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs 39289 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 39290	. 2 ⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))
2		relcnv 6094	. . . 4 ⊢ Rel ◡𝑅
3		df-funALTV 39267	. . . 4 ⊢ ( FunALTV ◡𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel ◡𝑅))
4	2, 3	mpbiran2 720	. . 3 ⊢ ( FunALTV ◡𝑅 ↔ CnvRefRel ≀ ◡𝑅)
5	4	anbi1i 633	. 2 ⊢ (( FunALTV ◡𝑅 ∧ Rel 𝑅) ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))
6	1, 5	bitr4i 280	1 ⊢ ( Disj 𝑅 ↔ ( FunALTV ◡𝑅 ∧ Rel 𝑅))

Syntax hints:   ↔ wb 208   ∧ wa 399  ^◡ccnv 5647  Rel wrel 5653   ≀ ccoss 38683   CnvRefRel wcnvrefrel 38692   FunALTV wfunALTV 38716   Disj wdisjALTV 38719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1564  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-ss 3922  df-opab 5164  df-xp 5654  df-rel 5655  df-cnv 5656  df-funALTV 39267  df-disjALTV 39290

This theorem is referenced by:  disjqmap2  39326  disjss  39331