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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 38039) 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 38159 | . 2 ⊢ redund ((( I ↾ dom 𝑅) ⊆ 𝑅 ∧ Rel 𝑅), RefRel 𝑅, ( RefRel 𝑅 ∧ SymRel 𝑅)) | |
2 | df-eqvrel 38109 | . . . 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 216 | . . 3 ⊢ ( EqvRel 𝑅 → ( RefRel 𝑅 ∧ SymRel 𝑅)) |
5 | 4 | redundpim3 38154 | . 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 394 ∧ w3a 1084 ⊆ wss 3941 I cid 5570 dom cdm 5673 ↾ cres 5675 Rel wrel 5678 RefRel wrefrel 37707 SymRel wsymrel 37713 TrRel wtrrel 37716 EqvRel weqvrel 37718 redund wredundp 37723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-br 5145 df-opab 5207 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-dm 5683 df-rn 5684 df-res 5685 df-refrel 38036 df-symrel 38068 df-eqvrel 38109 df-redundp 38149 |
This theorem is referenced by: refrelredund3 38161 |
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