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Mirrors > Home > MPE Home > Th. List > isinito | Structured version Visualization version GIF version |
Description: The predicate "is an initial 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 |
---|---|
isinito | β’ (π β (πΌ β (InitOβπΆ) β βπ β π΅ β!β β β (πΌπ»π))) |
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 | initoval 17942 | . . 3 β’ (π β (InitOβπΆ) = {π β π΅ β£ βπ β π΅ β!β β β (ππ»π)}) |
5 | 4 | eleq2d 2811 | . 2 β’ (π β (πΌ β (InitOβπΆ) 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 | oveq1 7408 | . . . . . . 7 β’ (π = πΌ β (ππ»π) = (πΌπ»π)) | |
8 | 7 | eleq2d 2811 | . . . . . 6 β’ (π = πΌ β (β β (ππ»π) β β β (πΌπ»π))) |
9 | 8 | eubidv 2572 | . . . . 5 β’ (π = πΌ β (β!β β β (ππ»π) β β!β β β (πΌπ»π))) |
10 | 9 | ralbidv 3169 | . . . 4 β’ (π = πΌ β (βπ β π΅ β!β β β (ππ»π) β βπ β π΅ β!β β β (πΌπ»π))) |
11 | 10 | elrab3 3676 | . . 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 (πΌπ»π))) |
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 (πΌπ»π))) |
13 | 5, 12 | bitrd 279 | 1 β’ (π β (πΌ β (InitOβπΆ) β βπ β π΅ β!β β β (πΌπ»π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 β!weu 2554 βwral 3053 {crab 3424 βcfv 6533 (class class class)co 7401 Basecbs 17140 Hom chom 17204 Catccat 17604 InitOcinito 17930 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-iota 6485 df-fun 6535 df-fv 6541 df-ov 7404 df-inito 17933 |
This theorem is referenced by: isinitoi 17948 initoeu2 17965 zrinitorngc 20523 zrninitoringc 20557 irinitoringc 21329 |
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