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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 4004 | . 2 ⊢ ( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I )) | |
2 | reli 5832 | . 2 ⊢ Rel I | |
3 | df-refrel 38016 | . 2 ⊢ ( RefRel I ↔ (( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I )) ∧ Rel I )) | |
4 | 1, 2, 3 | mpbir2an 709 | 1 ⊢ RefRel I |
Colors of variables: wff setvar class |
Syntax hints: ∩ cin 3948 ⊆ wss 3949 I cid 5579 × cxp 5680 dom cdm 5682 ran crn 5683 Rel wrel 5687 RefRel wrefrel 37687 |
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 2699 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3475 df-in 3956 df-ss 3966 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-refrel 38016 |
This theorem is referenced by: (None) |
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