Theorem algi 15074

Description: "Axiomatic" properties of Alg.

Hypotheses

Ref	Expression
algi.1	|- D = (dom` T)
algi.2	|- C = (cod` T)
algi.3	|- J = (id` T)
algi.4	|- R = (o` T)
algi.5	|- M = dom D
algi.6	|- O = dom J

Assertion

Ref	Expression
algi	|- (T e. Alg -> ((D:M-->O /\ C:M-->O /\ J:O-->M) /\ (Fun R /\ dom R C_ (M X. M) /\ ran R C_ M)))

Proof of Theorem algi

Step	Hyp	Ref	Expression
1		df-alg 15063	. . 3 |- Alg = {x | E.dE.cE.jE.r(x = <.<.d, c>., <.j, r>.>. /\ (d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d))}
2	1	eleq2i 1961	. 2 |- (T e. Alg <-> T e. {x | E.dE.cE.jE.r(x = <.<.d, c>., <.j, r>.>. /\ (d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d))})
3		3anass 862	. . . . . . . 8 |- ((x = <.<.d, c>., <.j, r>.>. /\ (d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d)) <-> (x = <.<.d, c>., <.j, r>.>. /\ ((d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d))))
4	3	bicomi 189	. . . . . . 7 |- ((x = <.<.d, c>., <.j, r>.>. /\ ((d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d))) <-> (x = <.<.d, c>., <.j, r>.>. /\ (d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d)))
5	4	exbii 1398	. . . . . 6 |- (E.r(x = <.<.d, c>., <.j, r>.>. /\ ((d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d))) <-> E.r(x = <.<.d, c>., <.j, r>.>. /\ (d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d)))
6	5	3exbii 1400	. . . . 5 |- (E.dE.cE.jE.r(x = <.<.d, c>., <.j, r>.>. /\ ((d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d))) <-> E.dE.cE.jE.r(x = <.<.d, c>., <.j, r>.>. /\ (d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d)))
7	6	abbii 2006	. . . 4 |- {x | E.dE.cE.jE.r(x = <.<.d, c>., <.j, r>.>. /\ ((d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d)))} = {x | E.dE.cE.jE.r(x = <.<.d, c>., <.j, r>.>. /\ (d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d))}
8	7	eleq2i 1961	. . 3 |- (T e. {x | E.dE.cE.jE.r(x = <.<.d, c>., <.j, r>.>. /\ ((d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d)))} <-> T e. {x | E.dE.cE.jE.r(x = <.<.d, c>., <.j, r>.>. /\ (d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d))})
9		algi.1	. . . . . . . 8 |- D = (dom` T)
10	9	domval 15070	. . . . . . 7 |- D = (1st` (1st` T))
11	10	eqcomi 1888	. . . . . 6 |- (1st` (1st` T)) = D
12	11	eqeq2i 1894	. . . . 5 |- (d = (1st` (1st` T)) <-> d = D)
13		feq1 4551	. . . . . . . 8 |- (d = D -> (d:dom d-->dom j <-> D:dom d-->dom j))
14		dmeq 4157	. . . . . . . . . 10 |- (d = D -> dom d = dom D)
15	14	feq2d 4557	. . . . . . . . 9 |- (d = D -> (D:dom d-->dom j <-> D:dom D-->dom j))
16		algi.5	. . . . . . . . . 10 |- M = dom D
17	16	feq2i 4559	. . . . . . . . 9 |- (D:M-->dom j <-> D:dom D-->dom j)
18	15, 17	syl6bbr 597	. . . . . . . 8 |- (d = D -> (D:dom d-->dom j <-> D:M-->dom j))
19	13, 18	bitrd 587	. . . . . . 7 |- (d = D -> (d:dom d-->dom j <-> D:M-->dom j))
20		feq2 4552	. . . . . . . . 9 |- (dom d = dom D -> (c:dom d-->dom j <-> c:dom D-->dom j))
21	16	feq2i 4559	. . . . . . . . 9 |- (c:M-->dom j <-> c:dom D-->dom j)
22	20, 21	syl6bbr 597	. . . . . . . 8 |- (dom d = dom D -> (c:dom d-->dom j <-> c:M-->dom j))
23	14, 22	syl 12	. . . . . . 7 |- (d = D -> (c:dom d-->dom j <-> c:M-->dom j))
24	16	eqcomi 1888	. . . . . . . . . 10 |- dom D = M
25	24	eqeq2i 1894	. . . . . . . . 9 |- (dom d = dom D <-> dom d = M)
26	25	biimpi 168	. . . . . . . 8 |- (dom d = dom D -> dom d = M)
27		feq3 4553	. . . . . . . 8 |- (dom d = M -> (j:dom j-->dom d <-> j:dom j-->M))
28	14, 26, 27	3syl 24	. . . . . . 7 |- (d = D -> (j:dom j-->dom d <-> j:dom j-->M))
29	19, 23, 28	3anbi123d 1168	. . . . . 6 |- (d = D -> ((d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) <-> (D:M-->dom j /\ c:M-->dom j /\ j:dom j-->M)))
30		xpid11 4181	. . . . . . . . 9 |- ((dom d X. dom d) = (M X. M) <-> dom d = M)
31		sseq2 2639	. . . . . . . . 9 |- ((dom d X. dom d) = (M X. M) -> (dom r C_ (dom d X. dom d) <-> dom r C_ (M X. M)))
32	30, 31	sylbir 218	. . . . . . . 8 |- (dom d = M -> (dom r C_ (dom d X. dom d) <-> dom r C_ (M X. M)))
33		sseq2 2639	. . . . . . . 8 |- (dom d = M -> (ran r C_ dom d <-> ran r C_ M))
34	32, 33	3anbi23d 1171	. . . . . . 7 |- (dom d = M -> ((Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d) <-> (Fun r /\ dom r C_ (M X. M) /\ ran r C_ M)))
35	14, 26, 34	3syl 24	. . . . . 6 |- (d = D -> ((Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d) <-> (Fun r /\ dom r C_ (M X. M) /\ ran r C_ M)))
36	29, 35	anbi12d 690	. . . . 5 |- (d = D -> (((d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d)) <-> ((D:M-->dom j /\ c:M-->dom j /\ j:dom j-->M) /\ (Fun r /\ dom r C_ (M X. M) /\ ran r C_ M))))
37	12, 36	sylbi 216	. . . 4 |- (d = (1st` (1st` T)) -> (((d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d)) <-> ((D:M-->dom j /\ c:M-->dom j /\ j:dom j-->M) /\ (Fun r /\ dom r C_ (M X. M) /\ ran r C_ M))))
38		algi.2	. . . . . . . 8 |- C = (cod` T)
39	38	codval 15071	. . . . . . 7 |- C = (2nd` (1st` T))
40	39	eqcomi 1888	. . . . . 6 |- (2nd` (1st` T)) = C
41	40	eqeq2i 1894	. . . . 5 |- (c = (2nd` (1st` T)) <-> c = C)
42		feq1 4551	. . . . . . 7 |- (c = C -> (c:M-->dom j <-> C:M-->dom j))
43	42	3anbi2d 1173	. . . . . 6 |- (c = C -> ((D:M-->dom j /\ c:M-->dom j /\ j:dom j-->M) <-> (D:M-->dom j /\ C:M-->dom j /\ j:dom j-->M)))
44	43	anbi1d 679	. . . . 5 |- (c = C -> (((D:M-->dom j /\ c:M-->dom j /\ j:dom j-->M) /\ (Fun r /\ dom r C_ (M X. M) /\ ran r C_ M)) <-> ((D:M-->dom j /\ C:M-->dom j /\ j:dom j-->M) /\ (Fun r /\ dom r C_ (M X. M) /\ ran r C_ M))))
45	41, 44	sylbi 216	. . . 4 |- (c = (2nd` (1st` T)) -> (((D:M-->dom j /\ c:M-->dom j /\ j:dom j-->M) /\ (Fun r /\ dom r C_ (M X. M) /\ ran r C_ M)) <-> ((D:M-->dom j /\ C:M-->dom j /\ j:dom j-->M) /\ (Fun r /\ dom r C_ (M X. M) /\ ran r C_ M))))
46		algi.3	. . . . . . . 8 |- J = (id` T)
47	46	idval 15072	. . . . . . 7 |- J = (1st` (2nd` T))
48	47	eqcomi 1888	. . . . . 6 |- (1st` (2nd` T)) = J
49	48	eqeq2i 1894	. . . . 5 |- (j = (1st` (2nd` T)) <-> j = J)
50		dmeq 4157	. . . . . . . 8 |- (j = J -> dom j = dom J)
51		algi.6	. . . . . . . . . . 11 |- O = dom J
52	51	eqcomi 1888	. . . . . . . . . 10 |- dom J = O
53	52	eqeq2i 1894	. . . . . . . . 9 |- (dom j = dom J <-> dom j = O)
54	53	biimpi 168	. . . . . . . 8 |- (dom j = dom J -> dom j = O)
55		feq3 4553	. . . . . . . 8 |- (dom j = O -> (D:M-->dom j <-> D:M-->O))
56	50, 54, 55	3syl 24	. . . . . . 7 |- (j = J -> (D:M-->dom j <-> D:M-->O))
57		feq3 4553	. . . . . . . 8 |- (dom j = O -> (C:M-->dom j <-> C:M-->O))
58	50, 54, 57	3syl 24	. . . . . . 7 |- (j = J -> (C:M-->dom j <-> C:M-->O))
59		feq1 4551	. . . . . . . 8 |- (j = J -> (j:dom j-->M <-> J:dom j-->M))
60		feq2 4552	. . . . . . . . 9 |- (dom j = O -> (J:dom j-->M <-> J:O-->M))
61	50, 54, 60	3syl 24	. . . . . . . 8 |- (j = J -> (J:dom j-->M <-> J:O-->M))
62	59, 61	bitrd 587	. . . . . . 7 |- (j = J -> (j:dom j-->M <-> J:O-->M))
63	56, 58, 62	3anbi123d 1168	. . . . . 6 |- (j = J -> ((D:M-->dom j /\ C:M-->dom j /\ j:dom j-->M) <-> (D:M-->O /\ C:M-->O /\ J:O-->M)))
64	63	anbi1d 679	. . . . 5 |- (j = J -> (((D:M-->dom j /\ C:M-->dom j /\ j:dom j-->M) /\ (Fun r /\ dom r C_ (M X. M) /\ ran r C_ M)) <-> ((D:M-->O /\ C:M-->O /\ J:O-->M) /\ (Fun r /\ dom r C_ (M X. M) /\ ran r C_ M))))
65	49, 64	sylbi 216	. . . 4 |- (j = (1st` (2nd` T)) -> (((D:M-->dom j /\ C:M-->dom j /\ j:dom j-->M) /\ (Fun r /\ dom r C_ (M X. M) /\ ran r C_ M)) <-> ((D:M-->O /\ C:M-->O /\ J:O-->M) /\ (Fun r /\ dom r C_ (M X. M) /\ ran r C_ M))))
66		algi.4	. . . . . . . 8 |- R = (o` T)
67	66	cmpval 15073	. . . . . . 7 |- R = (2nd` (2nd` T))
68	67	eqcomi 1888	. . . . . 6 |- (2nd` (2nd` T)) = R
69	68	eqeq2i 1894	. . . . 5 |- (r = (2nd` (2nd` T)) <-> r = R)
70		funeq 4441	. . . . . . 7 |- (r = R -> (Fun r <-> Fun R))
71		dmeq 4157	. . . . . . . 8 |- (r = R -> dom r = dom R)
72	71	sseq1d 2644	. . . . . . 7 |- (r = R -> (dom r C_ (M X. M) <-> dom R C_ (M X. M)))
73		rneq 4186	. . . . . . . 8 |- (r = R -> ran r = ran R)
74	73	sseq1d 2644	. . . . . . 7 |- (r = R -> (ran r C_ M <-> ran R C_ M))
75	70, 72, 74	3anbi123d 1168	. . . . . 6 |- (r = R -> ((Fun r /\ dom r C_ (M X. M) /\ ran r C_ M) <-> (Fun R /\ dom R C_ (M X. M) /\ ran R C_ M)))
76	75	anbi2d 678	. . . . 5 |- (r = R -> (((D:M-->O /\ C:M-->O /\ J:O-->M) /\ (Fun r /\ dom r C_ (M X. M) /\ ran r C_ M)) <-> ((D:M-->O /\ C:M-->O /\ J:O-->M) /\ (Fun R /\ dom R C_ (M X. M) /\ ran R C_ M))))
77	69, 76	sylbi 216	. . . 4 |- (r = (2nd` (2nd` T)) -> (((D:M-->O /\ C:M-->O /\ J:O-->M) /\ (Fun r /\ dom r C_ (M X. M) /\ ran r C_ M)) <-> ((D:M-->O /\ C:M-->O /\ J:O-->M) /\ (Fun R /\ dom R C_ (M X. M) /\ ran R C_ M))))
78	37, 45, 65, 77	eloi 14400	. . 3 |- (T e. {x | E.dE.cE.jE.r(x = <.<.d, c>., <.j, r>.>. /\ ((d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d)))} -> ((D:M-->O /\ C:M-->O /\ J:O-->M) /\ (Fun R /\ dom R C_ (M X. M) /\ ran R C_ M)))
79	8, 78	sylbir 218	. 2 |- (T e. {x | E.dE.cE.jE.r(x = <.<.d, c>., <.j, r>.>. /\ (d:dom d-->dom j /\ c:dom d-->dom j /\ j:dom j-->dom d) /\ (Fun r /\ dom r C_ (dom d X. dom d) /\ ran r C_ dom d))} -> ((D:M-->O /\ C:M-->O /\ J:O-->M) /\ (Fun R /\ dom R C_ (M X. M) /\ ran R C_ M)))
80	2, 79	sylbi 216	1 |- (T e. Alg -> ((D:M-->O /\ C:M-->O /\ J:O-->M) /\ (Fun R /\ dom R C_ (M X. M) /\ ran R C_ M)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   C_ wss 2593  <.cop 3046   X. cxp 3984  dom cdm 3986  ran crn 3987  Fun wfun 3992  -->wf 3994  ` cfv 3998  1stc1st 5018  2ndc2nd 5019   Alg calg 15058  domcdom_ 15059  codccod_ 15060  idcid_ 15061  oco_ 15062

This theorem is referenced by:  doma 15075  coda 15076  ida 15077  cmppfa 15079  dualalg 15131

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-13 1311  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  ax-un 3790

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-fo 4012  df-fv 4014  df-1st 5020  df-2nd 5021  df-alg 15063  df-doma 15064  df-coda 15065  df-ida 15066  df-cmpa 15067

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