Theorem 0alg 10660

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 2716	. . . 4 |- (/) e. V
2	1, 1, 1	3pm3.2i 820	. . 3 |- ((/) e. V /\ (/) e. V /\ (/) e. V)
3		dm0 3329	. . . . 5 |- dom (/) = (/)
4	3	eqcomi 1482	. . . 4 |- (/) = dom (/)
5	4, 4	isalg 10624	. . 3 |- ((((/) e. V /\ (/) e. V /\ (/) e. V) /\ (/) e. V) -> (<.<.(/), (/)>., <.(/), (/)>.>. e. Alg <-> (((/):(/)-->(/) /\ (/):(/)-->(/) /\ (/):(/)-->(/)) /\ (Fun (/) /\ dom (/) (_ ((/) X. (/)) /\ ran (/) (_ (/)))))
6	2, 1, 5	mp2an 699	. 2 |- (<.<.(/), (/)>., <.(/), (/)>.>. e. Alg <-> (((/):(/)-->(/) /\ (/):(/)-->(/) /\ (/):(/)-->(/)) /\ (Fun (/) /\ dom (/) (_ ((/) X. (/)) /\ ran (/) (_ (/))))
7		f0 3662	. . 3 |- (/):(/)-->(/)
8	7, 7, 7	3pm3.2i 820	. 2 |- ((/):(/)-->(/) /\ (/):(/)-->(/) /\ (/):(/)-->(/))
9		fun0 3550	. . 3 |- Fun (/)
10		ssid 2083	. . . 4 |- (/) (_ (/)
11		xp0r 3245	. . . 4 |- ((/) X. (/)) = (/)
12	10, 3, 11	3sstr4 2103	. . 3 |- dom (/) (_ ((/) X. (/))
13		rn0 3361	. . . 4 |- ran (/) = (/)
14	13	eqimssi 2114	. . 3 |- ran (/) (_ (/)
15	9, 12, 14	3pm3.2i 820	. 2 |- (Fun (/) /\ dom (/) (_ ((/) X. (/)) /\ ran (/) (_ (/))
16	6, 8, 15	mpbir2an 732	1 |- <.<.(/), (/)>., <.(/), (/)>.>. e. Alg

Syntax hints:   <-> wb 146   /\ wa 223   /\ w3a 777   e. wcel 960  Vcvv 1814   (_ wss 2050  (/)c0 2283  <.cop 2415   X. cxp 3174  dom cdm 3176  ran crn 3177  Fun wfun 3182  -->wf 3184  Algcalg 10614

This theorem is referenced by:  0ded 10661

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-fun 3198  df-fn 3199  df-f 3200  df-alg 10619

Copyright terms: Public domain