Theorem dfdisjALTV 36824

Description: Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs 36815 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 36816	. 2 ⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))
2		relcnv 6012	. . . 4 ⊢ Rel ◡𝑅
3		df-funALTV 36793	. . . 4 ⊢ ( FunALTV ◡𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel ◡𝑅))
4	2, 3	mpbiran2 707	. . 3 ⊢ ( FunALTV ◡𝑅 ↔ CnvRefRel ≀ ◡𝑅)
5	4	anbi1i 624	. 2 ⊢ (( FunALTV ◡𝑅 ∧ Rel 𝑅) ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))
6	1, 5	bitr4i 277	1 ⊢ ( Disj 𝑅 ↔ ( FunALTV ◡𝑅 ∧ Rel 𝑅))

Syntax hints:   ↔ wb 205   ∧ wa 396  ^◡ccnv 5588  Rel wrel 5594   ≀ ccoss 36333   CnvRefRel wcnvrefrel 36342   FunALTV wfunALTV 36364   Disj wdisjALTV 36367

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-funALTV 36793  df-disjALTV 36816

This theorem is referenced by:  disjss  36842