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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 22434 | . 2 ⊢ Ref = {〈𝑥, 𝑦〉 ∣ (∪ 𝑦 = ∪ 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∃𝑤 ∈ 𝑦 𝑧 ⊆ 𝑤)} | |
2 | 1 | relopabiv 5708 | 1 ⊢ Rel Ref |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∀wral 3064 ∃wrex 3065 ⊆ wss 3883 ∪ cuni 4836 Rel wrel 5574 Refcref 22431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-v 3425 df-in 3890 df-ss 3900 df-opab 5133 df-xp 5575 df-rel 5576 df-ref 22434 |
This theorem is referenced by: isref 22438 refbas 22439 refssex 22440 reftr 22443 refun0 22444 locfinref 31537 refssfne 34318 |
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