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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 17935 | . 2 β’ TermO = (π β Cat β¦ {π β (Baseβπ) β£ βπ β (Baseβπ)β!β β β (π(Hom βπ)π)}) | |
2 | fveq2 6892 | . . . 4 β’ (π = πΆ β (Baseβπ) = (BaseβπΆ)) | |
3 | initoval.b | . . . 4 β’ π΅ = (BaseβπΆ) | |
4 | 2, 3 | eqtr4di 2791 | . . 3 β’ (π = πΆ β (Baseβπ) = π΅) |
5 | fveq2 6892 | . . . . . . . 8 β’ (π = πΆ β (Hom βπ) = (Hom βπΆ)) | |
6 | initoval.h | . . . . . . . 8 β’ π» = (Hom βπΆ) | |
7 | 5, 6 | eqtr4di 2791 | . . . . . . 7 β’ (π = πΆ β (Hom βπ) = π») |
8 | 7 | oveqd 7426 | . . . . . 6 β’ (π = πΆ β (π(Hom βπ)π) = (ππ»π)) |
9 | 8 | eleq2d 2820 | . . . . 5 β’ (π = πΆ β (β β (π(Hom βπ)π) β β β (ππ»π))) |
10 | 9 | eubidv 2581 | . . . 4 β’ (π = πΆ β (β!β β β (π(Hom βπ)π) β β!β β β (ππ»π))) |
11 | 4, 10 | raleqbidv 3343 | . . 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 3450 | . 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 6906 | . . . 4 β’ π΅ β V |
15 | 14 | rabex 5333 | . . 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 7022 | 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 1542 β wcel 2107 β!weu 2563 βwral 3062 {crab 3433 Vcvv 3475 βcfv 6544 (class class class)co 7409 Basecbs 17144 Hom chom 17208 Catccat 17608 TermOctermo 17932 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-termo 17935 |
This theorem is referenced by: istermo 17947 istermoi 17950 dfinito2 17953 |
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