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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 35915) 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 36030 | . 2 ⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅)) | |
2 | df-eqvrel 35980 | . . . 4 ⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅)) | |
3 | 3simpa 1145 | . . . 4 ⊢ (( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅) → ( RefRel 𝑅 ∧ SymRel 𝑅)) | |
4 | 2, 3 | sylbi 220 | . . 3 ⊢ ( EqvRel 𝑅 → ( RefRel 𝑅 ∧ SymRel 𝑅)) |
5 | 4 | redundpim3 36025 | . 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 399 ∧ w3a 1084 ⊆ wss 3881 I cid 5424 dom cdm 5519 ↾ cres 5521 Rel wrel 5524 RefRel wrefrel 35619 SymRel wsymrel 35625 TrRel wtrrel 35628 EqvRel weqvrel 35630 redund wredundp 35635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-refrel 35912 df-symrel 35940 df-eqvrel 35980 df-redundp 36020 |
This theorem is referenced by: refrelredund3 36032 |
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