Theorem 0cat 15105

Description: Category 0 has no object and no morphism.

Assertion

Ref	Expression
0cat	|- <.<.(/), (/)>., <.(/), (/)>.>. e. Cat

Proof of Theorem 0cat

Step	Hyp	Ref	Expression
1		0ex 3446	. . . 4 |- (/) e. _V
2	1, 1, 1	3pm3.2i 1048	. . 3 |- ((/) e. _V /\ (/) e. _V /\ (/) e. _V)
3		eqid 1884	. . . 4 |- dom (/) = dom (/)
4	3, 3	iscat 15101	. . 3 |- ((((/) e. _V /\ (/) e. _V /\ (/) e. _V) /\ (/) e. _V) -> (<.<.(/), (/)>., <.(/), (/)>.>. e. Cat <-> ((<.<.(/), (/)>., <.(/), (/)>.>. e. Ded /\ A.f e. dom (/)A.g e. dom (/)A.h e. dom (/)((((/)` h) = ((/)` g) /\ ((/)` g) = ((/)` f)) -> (h(/)(g(/)f)) = ((h(/)g)(/)f))) /\ (A.a e. dom (/)A.f e. dom (/)(((/)` f) = a -> (((/)` a)(/)f) = f) /\ A.a e. dom (/)A.f e. dom (/)(((/)` f) = a -> (f(/)((/)` a)) = f)))))
5	2, 1, 4	mp2an 761	. 2 |- (<.<.(/), (/)>., <.(/), (/)>.>. e. Cat <-> ((<.<.(/), (/)>., <.(/), (/)>.>. e. Ded /\ A.f e. dom (/)A.g e. dom (/)A.h e. dom (/)((((/)` h) = ((/)` g) /\ ((/)` g) = ((/)` f)) -> (h(/)(g(/)f)) = ((h(/)g)(/)f))) /\ (A.a e. dom (/)A.f e. dom (/)(((/)` f) = a -> (((/)` a)(/)f) = f) /\ A.a e. dom (/)A.f e. dom (/)(((/)` f) = a -> (f(/)((/)` a)) = f))))
6		0ded 15104	. . 3 |- <.<.(/), (/)>., <.(/), (/)>.>. e. Ded
7		dm0 4170	. . . . . 6 |- dom (/) = (/)
8	7	eleq2i 1961	. . . . 5 |- (f e. dom (/) <-> f e. (/))
9		noel 2879	. . . . . 6 |- -. f e. (/)
10	9	pm2.21i 93	. . . . 5 |- (f e. (/) -> A.g e. dom (/)A.h e. dom (/)((((/)` h) = ((/)` g) /\ ((/)` g) = ((/)` f)) -> (h(/)(g(/)f)) = ((h(/)g)(/)f)))
11	8, 10	sylbi 216	. . . 4 |- (f e. dom (/) -> A.g e. dom (/)A.h e. dom (/)((((/)` h) = ((/)` g) /\ ((/)` g) = ((/)` f)) -> (h(/)(g(/)f)) = ((h(/)g)(/)f)))
12	11	rgen 2159	. . 3 |- A.f e. dom (/)A.g e. dom (/)A.h e. dom (/)((((/)` h) = ((/)` g) /\ ((/)` g) = ((/)` f)) -> (h(/)(g(/)f)) = ((h(/)g)(/)f))
13	6, 12	pm3.2i 307	. 2 |- (<.<.(/), (/)>., <.(/), (/)>.>. e. Ded /\ A.f e. dom (/)A.g e. dom (/)A.h e. dom (/)((((/)` h) = ((/)` g) /\ ((/)` g) = ((/)` f)) -> (h(/)(g(/)f)) = ((h(/)g)(/)f)))
14	7	eleq2i 1961	. . . . 5 |- (a e. dom (/) <-> a e. (/))
15		noel 2879	. . . . . 6 |- -. a e. (/)
16	15	pm2.21i 93	. . . . 5 |- (a e. (/) -> A.f e. dom (/)(((/)` f) = a -> (((/)` a)(/)f) = f))
17	14, 16	sylbi 216	. . . 4 |- (a e. dom (/) -> A.f e. dom (/)(((/)` f) = a -> (((/)` a)(/)f) = f))
18	17	rgen 2159	. . 3 |- A.a e. dom (/)A.f e. dom (/)(((/)` f) = a -> (((/)` a)(/)f) = f)
19	15	pm2.21i 93	. . . . 5 |- (a e. (/) -> A.f e. dom (/)(((/)` f) = a -> (f(/)((/)` a)) = f))
20	14, 19	sylbi 216	. . . 4 |- (a e. dom (/) -> A.f e. dom (/)(((/)` f) = a -> (f(/)((/)` a)) = f))
21	20	rgen 2159	. . 3 |- A.a e. dom (/)A.f e. dom (/)(((/)` f) = a -> (f(/)((/)` a)) = f)
22	18, 21	pm3.2i 307	. 2 |- (A.a e. dom (/)A.f e. dom (/)(((/)` f) = a -> (((/)` a)(/)f) = f) /\ A.a e. dom (/)A.f e. dom (/)(((/)` f) = a -> (f(/)((/)` a)) = f))
23	5, 13, 22	mpbir2an 800	1 |- <.<.(/), (/)>., <.(/), (/)>.>. e. Cat

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  (/)c0 2875  <.cop 3046  dom cdm 3986  ` cfv 3998  (class class class)co 4884   Ded cded 15081   Cat ccat 15099

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-alg 15063  df-ded 15082  df-cat 15100

Copyright terms: Public domain