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Mirrors > Home > MPE Home > Th. List > refrel | Structured version Visualization version GIF version |
Description: Refinement is a relation. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.) |
Ref | Expression |
---|---|
refrel | ⊢ Rel Ref |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ref 23529 | . 2 ⊢ Ref = {〈𝑥, 𝑦〉 ∣ (∪ 𝑦 = ∪ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)} | |
2 | 1 | relopabiv 5833 | 1 ⊢ Rel Ref |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 ∪ cuni 4912 Rel wrel 5694 Refcref 23526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-opab 5211 df-xp 5695 df-rel 5696 df-ref 23529 |
This theorem is referenced by: isref 23533 refbas 23534 refssex 23535 reftr 23538 refun0 23539 locfinref 33802 refssfne 36341 |
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