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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 3989 | . 2 ⊢ ( I ∩ (dom I × ran I )) ⊆ ( I ∩ (dom I × ran I )) | |
2 | reli 5698 | . 2 ⊢ Rel I | |
3 | df-refrel 35767 | . 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 3935 ⊆ wss 3936 I cid 5459 × cxp 5553 dom cdm 5555 ran crn 5556 Rel wrel 5560 RefRel wrefrel 35474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-refrel 35767 |
This theorem is referenced by: (None) |
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