![]() |
Mathbox for Peter Mazsa |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > refrelid | Structured version Visualization version GIF version |
Description: Identity relation is reflexive. (Contributed by Peter Mazsa, 25-Jul-2021.) |
Ref | Expression |
---|---|
refrelid | ⊢ RefRel I |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3967 | . 2 ⊢ ( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I )) | |
2 | reli 5783 | . 2 ⊢ Rel I | |
3 | df-refrel 37020 | . 2 ⊢ ( RefRel I ↔ (( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I )) ∧ Rel I )) | |
4 | 1, 2, 3 | mpbir2an 710 | 1 ⊢ RefRel I |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3910 ⊆ wss 3911 I cid 5531 × cxp 5632 dom cdm 5634 ran crn 5635 Rel wrel 5639 RefRel wrefrel 36686 |
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-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-in 3918 df-ss 3928 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-refrel 37020 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |