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Mirrors > Home > MPE Home > Th. List > termoval | Structured version Visualization version GIF version |
Description: The value of the terminal object function, i.e. the set of all terminal objects of a category. (Contributed by AV, 3-Apr-2020.) |
Ref | Expression |
---|---|
initoval.c | β’ (π β πΆ β Cat) |
initoval.b | β’ π΅ = (BaseβπΆ) |
initoval.h | β’ π» = (Hom βπΆ) |
Ref | Expression |
---|---|
termoval | β’ (π β (TermOβπΆ) = {π β π΅ β£ βπ β π΅ β!β β β (ππ»π)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-termo 17937 | . 2 β’ TermO = (π β Cat β¦ {π β (Baseβπ) β£ βπ β (Baseβπ)β!β β β (π(Hom βπ)π)}) | |
2 | fveq2 6891 | . . . 4 β’ (π = πΆ β (Baseβπ) = (BaseβπΆ)) | |
3 | initoval.b | . . . 4 β’ π΅ = (BaseβπΆ) | |
4 | 2, 3 | eqtr4di 2790 | . . 3 β’ (π = πΆ β (Baseβπ) = π΅) |
5 | fveq2 6891 | . . . . . . . 8 β’ (π = πΆ β (Hom βπ) = (Hom βπΆ)) | |
6 | initoval.h | . . . . . . . 8 β’ π» = (Hom βπΆ) | |
7 | 5, 6 | eqtr4di 2790 | . . . . . . 7 β’ (π = πΆ β (Hom βπ) = π») |
8 | 7 | oveqd 7428 | . . . . . 6 β’ (π = πΆ β (π(Hom βπ)π) = (ππ»π)) |
9 | 8 | eleq2d 2819 | . . . . 5 β’ (π = πΆ β (β β (π(Hom βπ)π) β β β (ππ»π))) |
10 | 9 | eubidv 2580 | . . . 4 β’ (π = πΆ β (β!β β β (π(Hom βπ)π) β β!β β β (ππ»π))) |
11 | 4, 10 | raleqbidv 3342 | . . 3 β’ (π = πΆ β (βπ β (Baseβπ)β!β β β (π(Hom βπ)π) β βπ β π΅ β!β β β (ππ»π))) |
12 | 4, 11 | rabeqbidv 3449 | . 2 β’ (π = πΆ β {π β (Baseβπ) β£ βπ β (Baseβπ)β!β β β (π(Hom βπ)π)} = {π β π΅ β£ βπ β π΅ β!β β β (ππ»π)}) |
13 | initoval.c | . 2 β’ (π β πΆ β Cat) | |
14 | 3 | fvexi 6905 | . . . 4 β’ π΅ β V |
15 | 14 | rabex 5332 | . . 3 β’ {π β π΅ β£ βπ β π΅ β!β β β (ππ»π)} β V |
16 | 15 | a1i 11 | . 2 β’ (π β {π β π΅ β£ βπ β π΅ β!β β β (ππ»π)} β V) |
17 | 1, 12, 13, 16 | fvmptd3 7021 | 1 β’ (π β (TermOβπΆ) = {π β π΅ β£ βπ β π΅ β!β β β (ππ»π)}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 β!weu 2562 βwral 3061 {crab 3432 Vcvv 3474 βcfv 6543 (class class class)co 7411 Basecbs 17146 Hom chom 17210 Catccat 17610 TermOctermo 17934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7414 df-termo 17937 |
This theorem is referenced by: istermo 17949 istermoi 17952 dfinito2 17955 |
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