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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 37006) 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 37126 | . 2 ⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅)) | |
2 | df-eqvrel 37076 | . . . 4 ⊢ ( EqvRel 𝑅 ↔ ( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅)) | |
3 | 3simpa 1149 | . . . 4 ⊢ (( RefRel 𝑅 ∧ SymRel 𝑅 ∧ TrRel 𝑅) → ( RefRel 𝑅 ∧ SymRel 𝑅)) | |
4 | 2, 3 | sylbi 216 | . . 3 ⊢ ( EqvRel 𝑅 → ( RefRel 𝑅 ∧ SymRel 𝑅)) |
5 | 4 | redundpim3 37121 | . 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 397 ∧ w3a 1088 ⊆ wss 3915 I cid 5535 dom cdm 5638 ↾ cres 5640 Rel wrel 5643 RefRel wrefrel 36669 SymRel wsymrel 36675 TrRel wtrrel 36678 EqvRel weqvrel 36680 redund wredundp 36685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-dm 5648 df-rn 5649 df-res 5650 df-refrel 37003 df-symrel 37035 df-eqvrel 37076 df-redundp 37116 |
This theorem is referenced by: refrelredund3 37128 |
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