Theorem 0cat 10571

Description: Category 0. No object, no morphism.

Assertion

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

Proof of Theorem 0cat

Step	Hyp	Ref	Expression
1		0ex 2706	. . . 4 |- (/) e. V
2	1, 1, 1	3pm3.2i 817	. . 3 |- ((/) e. V /\ (/) e. V /\ (/) e. V)
3		eqid 1473	. . . 4 |- dom (/) = dom (/)
4	3, 3	iscat 10567	. . 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 696	. 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 10570	. . 3 |- <.<.(/), (/)>., <.(/), (/)>.>. e. Ded
7		dm0 3318	. . . . . 6 |- dom (/) = (/)
8	7	eleq2i 1535	. . . . 5 |- (f e. dom (/) <-> f e. (/))
9		noel 2280	. . . . . 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 1695	. . 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 1535	. . . . 5 |- (a e. dom (/) <-> a e. (/))
15		noel 2280	. . . . . 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 1695	. . 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 1695	. . 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 729	1 |- <.<.(/), (/)>., <.(/), (/)>.>. e. Cat

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  A.wral 1642  Vcvv 1807  (/)c0 2276  <.cop 2407  dom cdm 3165  ` cfv 3177  (class class class)co 3954  Dedcded 10547  Catccat 10565

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-opr 3956  df-alg 10528  df-ded 10548  df-cat 10566

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