Theorem aidm 10683

Description: The underlying directed multi graph of a deductive system.

Hypotheses

Ref	Expression
aidm.1	|- D = (dom` T)
aidm.2	|- C = (cod` T)
aidm.3	|- O = dom (id` T)

Assertion

Ref	Expression
aidm	|- (T e. Ded -> <.<.D, C>., ran D>. e. Dgra)

Proof of Theorem aidm

Step	Hyp	Ref	Expression
1		dedalg 10676	. . . . 5 |- (T e. Ded -> T e. Alg)
2		eqid 1475	. . . . . 6 |- dom D = dom D
3		aidm.1	. . . . . 6 |- D = (dom` T)
4		aidm.3	. . . . . 6 |- O = dom (id` T)
5		eqid 1475	. . . . . 6 |- (id` T) = (id` T)
6	2, 3, 4, 5	doma 10661	. . . . 5 |- (T e. Alg -> D:dom D-->O)
7	1, 6	syl 10	. . . 4 |- (T e. Ded -> D:dom D-->O)
8	4, 3	rdmob 10681	. . . . 5 |- (T e. Ded -> ran D = O)
9		feq3 3622	. . . . 5 |- (ran D = O -> (D:dom D-->ran D <-> D:dom D-->O))
10	8, 9	syl 10	. . . 4 |- (T e. Ded -> (D:dom D-->ran D <-> D:dom D-->O))
11	7, 10	mpbird 196	. . 3 |- (T e. Ded -> D:dom D-->ran D)
12	3	dmeqi 3312	. . . . . 6 |- dom D = dom (dom` T)
13		eqid 1475	. . . . . 6 |- (dom` T) = (dom` T)
14		eqid 1475	. . . . . 6 |- dom (id` T) = dom (id` T)
15		aidm.2	. . . . . 6 |- C = (cod` T)
16	12, 13, 14, 5, 15	coda 10662	. . . . 5 |- (T e. Alg -> C:dom D-->dom (id` T))
17	1, 16	syl 10	. . . 4 |- (T e. Ded -> C:dom D-->dom (id` T))
18	14, 3	rdmob 10681	. . . . 5 |- (T e. Ded -> ran D = dom (id` T))
19		feq3 3622	. . . . 5 |- (ran D = dom (id` T) -> (C:dom D-->ran D <-> C:dom D-->dom (id` T)))
20	18, 19	syl 10	. . . 4 |- (T e. Ded -> (C:dom D-->ran D <-> C:dom D-->dom (id` T)))
21	17, 20	mpbird 196	. . 3 |- (T e. Ded -> C:dom D-->ran D)
22	11, 21	jca 288	. 2 |- (T e. Ded -> (D:dom D-->ran D /\ C:dom D-->ran D))
23		ismgra 10642	. . 3 |- ((D e. V /\ C e. V /\ ran D e. V) -> (<.<.D, C>., ran D>. e. Dgra <-> (D:dom D-->ran D /\ C:dom D-->ran D)))
24	3	a1i 8	. . . 4 |- (T e. Ded -> D = (dom` T))
25		fvex 3732	. . . 4 |- (dom` T) e. V
26	24, 25	syl6eqel 1556	. . 3 |- (T e. Ded -> D e. V)
27	15	a1i 8	. . . 4 |- (T e. Ded -> C = (cod` T))
28		fvex 3732	. . . 4 |- (cod` T) e. V
29	27, 28	syl6eqel 1556	. . 3 |- (T e. Ded -> C e. V)
30	24	rneqd 3341	. . . 4 |- (T e. Ded -> ran D = ran (dom` T))
31	25	a1i 8	. . . . 5 |- (T e. Ded -> (dom` T) e. V)
32		rnexg 3359	. . . . 5 |- ((dom` T) e. V -> ran (dom` T) e. V)
33	31, 32	syl 10	. . . 4 |- (T e. Ded -> ran (dom` T) e. V)
34	30, 33	eqeltrd 1548	. . 3 |- (T e. Ded -> ran D e. V)
35	23, 26, 29, 34	syl3anc 858	. 2 |- (T e. Ded -> (<.<.D, C>., ran D>. e. Dgra <-> (D:dom D-->ran D /\ C:dom D-->ran D)))
36	22, 35	mpbird 196	1 |- (T e. Ded -> <.<.D, C>., ran D>. e. Dgra)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811  <.cop 2411  dom cdm 3170  ran crn 3171  -->wf 3178  ` cfv 3182  Dgracmgra 10640  Algcalg 10643  domcdom_ 10644  codccod_ 10645  idcid_ 10646  Dedcded 10667

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  ax-un 2866

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-rex 1650  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-int 2534  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-fo 3196  df-fv 3198  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-mgra 10641  df-alg 10648  df-doma 10649  df-coda 10650  df-ida 10651  df-cmpa 10652  df-ded 10668

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