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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfdisjALTV | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
dfdisjALTV | ⊢ ( Disj 𝑅 ↔ ( FunALTV ◡𝑅 ∧ Rel 𝑅)) |
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)) |
Colors of variables: wff setvar class |
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 |
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