Theorem dfdisjALTV 35980

Description: Alternate definition of the disjoint relation predicate. A disjoint relation is a converse function of the relation, see the comment of df-disjs 35971 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 35972	. 2 ⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅))
2		relcnv 5960	. . . 4 ⊢ Rel ◡𝑅
3		df-funALTV 35949	. . . 4 ⊢ ( FunALTV ◡𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel ◡𝑅))
4	2, 3	mpbiran2 708	. . 3 ⊢ ( FunALTV ◡𝑅 ↔ CnvRefRel ≀ ◡𝑅)
5	4	anbi1i 625	. 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 398  ^◡ccnv 5547  Rel wrel 5553   ≀ ccoss 35487   CnvRefRel wcnvrefrel 35496   FunALTV wfunALTV 35518   Disj wdisjALTV 35521

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-rab 3146  df-v 3493  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-nul 4285  df-if 4461  df-sn 4561  df-pr 4563  df-op 4567  df-opab 5122  df-xp 5554  df-rel 5555  df-cnv 5556  df-funALTV 35949  df-disjALTV 35972

This theorem is referenced by:  disjss  35998