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Mirrors > Home > MPE Home > Th. List > Mathboxes > reflexg | Structured version Visualization version GIF version |
Description: Two ways of saying a relation is reflexive over its domain and range. (Contributed by RP, 4-Aug-2020.) |
Ref | Expression |
---|---|
reflexg | ⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | undmrnresiss 39984 | 1 ⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 ∪ cun 3934 ⊆ wss 3936 class class class wbr 5066 I cid 5459 dom cdm 5555 ran crn 5556 ↾ cres 5557 |
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 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
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-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 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-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 |
This theorem is referenced by: refimssco 39987 |
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