Theorem 0cat 10691

Description: Category 0. No object, no morphism.

Assertion

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

Proof of Theorem 0cat

Step	Hyp	Ref	Expression
1		0ex 2711	. . . 4 |- (/) e. V
2	1, 1, 1	3pm3.2i 818	. . 3 |- ((/) e. V /\ (/) e. V /\ (/) e. V)
3		eqid 1475	. . . 4 |- dom (/) = dom (/)
4	3, 3	iscat 10687	. . 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 697	. 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 10690	. . 3 |- <.<.(/), (/)>., <.(/), (/)>.>. e. Ded
7		dm0 3323	. . . . . 6 |- dom (/) = (/)
8	7	eleq2i 1538	. . . . 5 |- (f e. dom (/) <-> f e. (/))
9		noel 2284	. . . . . 6 |- -. f e. (/)
10	9	pm2.21i 77	. . . . 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 199	. . . 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 1698	. . 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 285	. 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 1538	. . . . 5 |- (a e. dom (/) <-> a e. (/))
15		noel 2284	. . . . . 6 |- -. a e. (/)
16	15	pm2.21i 77	. . . . 5 |- (a e. (/) -> A.f e. dom (/)(((/)` f) = a -> (((/)` a)(/)f) = f))
17	14, 16	sylbi 199	. . . 4 |- (a e. dom (/) -> A.f e. dom (/)(((/)` f) = a -> (((/)` a)(/)f) = f))
18	17	rgen 1698	. . 3 |- A.a e. dom (/)A.f e. dom (/)(((/)` f) = a -> (((/)` a)(/)f) = f)
19	15	pm2.21i 77	. . . . 5 |- (a e. (/) -> A.f e. dom (/)(((/)` f) = a -> (f(/)((/)` a)) = f))
20	14, 19	sylbi 199	. . . 4 |- (a e. dom (/) -> A.f e. dom (/)(((/)` f) = a -> (f(/)((/)` a)) = f))
21	20	rgen 1698	. . 3 |- A.a e. dom (/)A.f e. dom (/)(((/)` f) = a -> (f(/)((/)` a)) = f)
22	18, 21	pm3.2i 285	. 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 730	1 |- <.<.(/), (/)>., <.(/), (/)>.>. e. Cat

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811  (/)c0 2280  <.cop 2411  dom cdm 3170  ` cfv 3182  (class class class)co 3963  Dedcded 10667  Catccat 10685

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-alg 10648  df-ded 10668  df-cat 10686

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