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Mirrors > Home > MPE Home > Th. List > istermo | Structured version Visualization version GIF version |
Description: The predicate "is a terminal object" of a category. (Contributed by AV, 3-Apr-2020.) |
Ref | Expression |
---|---|
isinito.b | β’ π΅ = (BaseβπΆ) |
isinito.h | β’ π» = (Hom βπΆ) |
isinito.c | β’ (π β πΆ β Cat) |
isinito.i | β’ (π β πΌ β π΅) |
Ref | Expression |
---|---|
istermo | β’ (π β (πΌ β (TermOβπΆ) β βπ β π΅ β!β β β (ππ»πΌ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isinito.c | . . . 4 β’ (π β πΆ β Cat) | |
2 | isinito.b | . . . 4 β’ π΅ = (BaseβπΆ) | |
3 | isinito.h | . . . 4 β’ π» = (Hom βπΆ) | |
4 | 1, 2, 3 | termoval 17953 | . . 3 β’ (π β (TermOβπΆ) = {π β π΅ β£ βπ β π΅ β!β β β (ππ»π)}) |
5 | 4 | eleq2d 2813 | . 2 β’ (π β (πΌ β (TermOβπΆ) β πΌ β {π β π΅ β£ βπ β π΅ β!β β β (ππ»π)})) |
6 | isinito.i | . . 3 β’ (π β πΌ β π΅) | |
7 | oveq2 7412 | . . . . . . 7 β’ (π = πΌ β (ππ»π) = (ππ»πΌ)) | |
8 | 7 | eleq2d 2813 | . . . . . 6 β’ (π = πΌ β (β β (ππ»π) β β β (ππ»πΌ))) |
9 | 8 | eubidv 2574 | . . . . 5 β’ (π = πΌ β (β!β β β (ππ»π) β β!β β β (ππ»πΌ))) |
10 | 9 | ralbidv 3171 | . . . 4 β’ (π = πΌ β (βπ β π΅ β!β β β (ππ»π) β βπ β π΅ β!β β β (ππ»πΌ))) |
11 | 10 | elrab3 3679 | . . 3 β’ (πΌ β π΅ β (πΌ β {π β π΅ β£ βπ β π΅ β!β β β (ππ»π)} β βπ β π΅ β!β β β (ππ»πΌ))) |
12 | 6, 11 | syl 17 | . 2 β’ (π β (πΌ β {π β π΅ β£ βπ β π΅ β!β β β (ππ»π)} β βπ β π΅ β!β β β (ππ»πΌ))) |
13 | 5, 12 | bitrd 279 | 1 β’ (π β (πΌ β (TermOβπΆ) β βπ β π΅ β!β β β (ππ»πΌ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 β!weu 2556 βwral 3055 {crab 3426 βcfv 6536 (class class class)co 7404 Basecbs 17150 Hom chom 17214 Catccat 17614 TermOctermo 17941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-ov 7407 df-termo 17944 |
This theorem is referenced by: istermoi 17959 zrtermorngc 20536 zrtermoringc 20568 |
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