Theorem 0alg 10689

Description: An "algebra" with no object and no morphism.

Assertion

Ref	Expression
0alg	|- <.<.(/), (/)>., <.(/), (/)>.>. e. Alg

Proof of Theorem 0alg

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		dm0 3323	. . . . 5 |- dom (/) = (/)
4	3	eqcomi 1479	. . . 4 |- (/) = dom (/)
5	4, 4	isalg 10653	. . 3 |- ((((/) e. V /\ (/) e. V /\ (/) e. V) /\ (/) e. V) -> (<.<.(/), (/)>., <.(/), (/)>.>. e. Alg <-> (((/):(/)-->(/) /\ (/):(/)-->(/) /\ (/):(/)-->(/)) /\ (Fun (/) /\ dom (/) (_ ((/) X. (/)) /\ ran (/) (_ (/)))))
6	2, 1, 5	mp2an 697	. 2 |- (<.<.(/), (/)>., <.(/), (/)>.>. e. Alg <-> (((/):(/)-->(/) /\ (/):(/)-->(/) /\ (/):(/)-->(/)) /\ (Fun (/) /\ dom (/) (_ ((/) X. (/)) /\ ran (/) (_ (/))))
7		f0 3656	. . 3 |- (/):(/)-->(/)
8	7, 7, 7	3pm3.2i 818	. 2 |- ((/):(/)-->(/) /\ (/):(/)-->(/) /\ (/):(/)-->(/))
9		fun0 3544	. . 3 |- Fun (/)
10		ssid 2080	. . . 4 |- (/) (_ (/)
11		xp0r 3239	. . . 4 |- ((/) X. (/)) = (/)
12	10, 3, 11	3sstr4 2100	. . 3 |- dom (/) (_ ((/) X. (/))
13		rn0 3355	. . . 4 |- ran (/) = (/)
14	13	eqimssi 2111	. . 3 |- ran (/) (_ (/)
15	9, 12, 14	3pm3.2i 818	. 2 |- (Fun (/) /\ dom (/) (_ ((/) X. (/)) /\ ran (/) (_ (/))
16	6, 8, 15	mpbir2an 730	1 |- <.<.(/), (/)>., <.(/), (/)>.>. e. Alg

Syntax hints:   <-> wb 146   /\ wa 223   /\ w3a 775   e. wcel 958  Vcvv 1811   (_ wss 2047  (/)c0 2280  <.cop 2411   X. cxp 3168  dom cdm 3170  ran crn 3171  Fun wfun 3176  -->wf 3178  Algcalg 10643

This theorem is referenced by:  0ded 10690

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-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-fun 3192  df-fn 3193  df-f 3194  df-alg 10648

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