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Mathbox for Peter Mazsa |
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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 37877 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 37878 | . 2 ⊢ ( Disj 𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅)) | |
2 | relcnv 6102 | . . . 4 ⊢ Rel ◡𝑅 | |
3 | df-funALTV 37855 | . . . 4 ⊢ ( FunALTV ◡𝑅 ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel ◡𝑅)) | |
4 | 2, 3 | mpbiran2 706 | . . 3 ⊢ ( FunALTV ◡𝑅 ↔ CnvRefRel ≀ ◡𝑅) |
5 | 4 | anbi1i 622 | . 2 ⊢ (( FunALTV ◡𝑅 ∧ Rel 𝑅) ↔ ( CnvRefRel ≀ ◡𝑅 ∧ Rel 𝑅)) |
6 | 1, 5 | bitr4i 277 | 1 ⊢ ( Disj 𝑅 ↔ ( FunALTV ◡𝑅 ∧ Rel 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 ◡ccnv 5674 Rel wrel 5680 ≀ ccoss 37346 CnvRefRel wcnvrefrel 37355 FunALTV wfunALTV 37377 Disj wdisjALTV 37380 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-v 3474 df-in 3954 df-ss 3964 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 df-funALTV 37855 df-disjALTV 37878 |
This theorem is referenced by: disjss 37904 |
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