Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > refrelredund2 | Structured version Visualization version GIF version |
Description: The naive version of the definition of reflexive relation is redundant with respect to reflexive relation (see dfrefrel2 35789) in equivalence relation. (Contributed by Peter Mazsa, 25-Oct-2022.) |
Ref | Expression |
---|---|
refrelredund2 | ⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | refrelredund4 35904 | . 2 ⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅)) | |
2 | df-eqvrel 35854 | . . . 4 ⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅)) | |
3 | 3simpa 1143 | . . . 4 ⊢ (( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅) → ( RefRel 𝑅 ∧ SymRel 𝑅)) | |
4 | 2, 3 | sylbi 219 | . . 3 ⊢ ( EqvRel 𝑅 → ( RefRel 𝑅 ∧ SymRel 𝑅)) |
5 | 4 | redundpim3 35899 | . 2 ⊢ ( redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅)) → redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅)) |
6 | 1, 5 | ax-mp 5 | 1 ⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, EqvRel 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∧ w3a 1082 ⊆ wss 3929 I cid 5452 dom cdm 5548 ↾ cres 5550 Rel wrel 5553 RefRel wrefrel 35493 SymRel wsymrel 35499 TrRel wtrrel 35502 EqvRel weqvrel 35504 redund wredundp 35509 |
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 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
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-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 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-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-refrel 35786 df-symrel 35814 df-eqvrel 35854 df-redundp 35894 |
This theorem is referenced by: refrelredund3 35906 |
Copyright terms: Public domain | W3C validator |