Theorem dfdisjALTV 39731

Description: Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs 39722 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 39723	. 2 ⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))
2		relcnv 6099	. . . 4 ⊢ Rel ◡𝑅
3		df-funALTV 39700	. . . 4 ⊢ ( FunALTV ◡𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel ◡𝑅))
4	2, 3	mpbiran2 723	. . 3 ⊢ ( FunALTV ◡𝑅 ↔ CnvRefRel ≀ ◡𝑅)
5	4	anbi1i 636	. 2 ⊢ (( FunALTV ◡𝑅 ∧ Rel 𝑅) ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))
6	1, 5	bitr4i 281	1 ⊢ ( Disj 𝑅 ↔ ( FunALTV ◡𝑅 ∧ Rel 𝑅))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401  ^◡ccnv 5649  Rel wrel 5655   ≀ ccoss 39116   CnvRefRel wcnvrefrel 39125   FunALTV wfunALTV 39149   Disj wdisjALTV 39152

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-funALTV 39700  df-disjALTV 39723

This theorem is used by:  disjqmap2  39759  disjss  39764