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Mathbox for Rohan Ridenour |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > scotteqd | Structured version Visualization version GIF version |
Description: Equality theorem for the Scott operation. (Contributed by Rohan Ridenour, 9-Aug-2023.) |
Ref | Expression |
---|---|
scotteqd.1 | β’ (π β π΄ = π΅) |
Ref | Expression |
---|---|
scotteqd | β’ (π β Scott π΄ = Scott π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scotteqd.1 | . . 3 β’ (π β π΄ = π΅) | |
2 | 1 | adantr 482 | . . . 4 β’ ((π β§ π₯ β π΄) β π΄ = π΅) |
3 | 2 | raleqdv 3312 | . . 3 β’ ((π β§ π₯ β π΄) β (βπ¦ β π΄ (rankβπ₯) β (rankβπ¦) β βπ¦ β π΅ (rankβπ₯) β (rankβπ¦))) |
4 | 1, 3 | rabeqbidva 3422 | . 2 β’ (π β {π₯ β π΄ β£ βπ¦ β π΄ (rankβπ₯) β (rankβπ¦)} = {π₯ β π΅ β£ βπ¦ β π΅ (rankβπ₯) β (rankβπ¦)}) |
5 | df-scott 42608 | . 2 β’ Scott π΄ = {π₯ β π΄ β£ βπ¦ β π΄ (rankβπ₯) β (rankβπ¦)} | |
6 | df-scott 42608 | . 2 β’ Scott π΅ = {π₯ β π΅ β£ βπ¦ β π΅ (rankβπ₯) β (rankβπ¦)} | |
7 | 4, 5, 6 | 3eqtr4g 2798 | 1 β’ (π β Scott π΄ = Scott π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3061 {crab 3406 β wss 3914 βcfv 6500 rankcrnk 9707 Scott cscott 42607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rab 3407 df-scott 42608 |
This theorem is referenced by: scotteq 42610 dfcoll2 42624 colleq12d 42625 |
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