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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 43593 | 1 ⊢ (( I ↾ (dom 𝐴 ∪ ran 𝐴)) ⊆ 𝐴 ↔ ∀𝑥∀𝑦(𝑥𝐴𝑦 → (𝑥𝐴𝑥 ∧ 𝑦𝐴𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∀wal 1534 ∪ cun 3960 ⊆ wss 3962 class class class wbr 5147 I cid 5581 dom cdm 5688 ran crn 5689 ↾ cres 5690 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-11 2154 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 |
This theorem is referenced by: refimssco 43596 |
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