Theorem rdmob 15095

Description: The range of (dom` T) is the class of the objects.

Hypotheses

Ref	Expression
rdmob.1	|- O = dom (id` T)
rdmob.2	|- D = (dom` T)

Assertion

Ref	Expression
rdmob	|- (T e. Ded -> ran D = O)

Proof of Theorem rdmob

Step	Hyp	Ref	Expression
1		dedalg 15090	. . 3 |- (T e. Ded -> T e. Alg )
2		eqid 1884	. . . 4 |- dom D = dom D
3		rdmob.2	. . . 4 |- D = (dom` T)
4		rdmob.1	. . . 4 |- O = dom (id` T)
5		eqid 1884	. . . 4 |- (id` T) = (id` T)
6	2, 3, 4, 5	doma 15075	. . 3 |- (T e. Alg -> D:dom D-->O)
7		frn 4569	. . 3 |- (D:dom D-->O -> ran D C_ O)
8	1, 6, 7	3syl 24	. 2 |- (T e. Ded -> ran D C_ O)
9		eqid 1884	. . . . . 6 |- (cod` T) = (cod` T)
10	4, 3, 5, 9	idosd 15091	. . . . 5 |- ((T e. Ded /\ a e. O) -> ((D` ((id` T)` a)) = a /\ ((cod` T)` ((id` T)` a)) = a))
11		eqid 1884	. . . . . . . . 9 |- dom (dom` T) = dom (dom` T)
12		eqid 1884	. . . . . . . . 9 |- (dom` T) = (dom` T)
13	11, 12, 4, 5	ida 15077	. . . . . . . 8 |- (T e. Alg -> (id` T):O-->dom (dom` T))
14		ffvelrn 4787	. . . . . . . . 9 |- (((id` T):O-->dom (dom` T) /\ a e. O) -> ((id` T)` a) e. dom (dom` T))
15	14	ex 402	. . . . . . . 8 |- ((id` T):O-->dom (dom` T) -> (a e. O -> ((id` T)` a) e. dom (dom` T)))
16	1, 13, 15	3syl 24	. . . . . . 7 |- (T e. Ded -> (a e. O -> ((id` T)` a) e. dom (dom` T)))
17		eqid 1884	. . . . . . . . 9 |- dom (id` T) = dom (id` T)
18	11, 12, 17, 5	doma 15075	. . . . . . . 8 |- (T e. Alg -> (dom` T):dom (dom` T)-->dom (id` T))
19		ffun 4565	. . . . . . . . . 10 |- ((dom` T):dom (dom` T)-->dom (id` T) -> Fun (dom` T))
20		fvelrn 4785	. . . . . . . . . . 11 |- ((Fun (dom` T) /\ ((id` T)` a) e. dom (dom` T)) -> ((dom` T)` ((id` T)` a)) e. ran (dom` T))
21	20	ex 402	. . . . . . . . . 10 |- (Fun (dom` T) -> (((id` T)` a) e. dom (dom` T) -> ((dom` T)` ((id` T)` a)) e. ran (dom` T)))
22	19, 21	syl 12	. . . . . . . . 9 |- ((dom` T):dom (dom` T)-->dom (id` T) -> (((id` T)` a) e. dom (dom` T) -> ((dom` T)` ((id` T)` a)) e. ran (dom` T)))
23	3	fveq1i 4682	. . . . . . . . . 10 |- (D` ((id` T)` a)) = ((dom` T)` ((id` T)` a))
24	23	eleq1i 1960	. . . . . . . . 9 |- ((D` ((id` T)` a)) e. ran (dom` T) <-> ((dom` T)` ((id` T)` a)) e. ran (dom` T))
25	22, 24	syl6ibr 230	. . . . . . . 8 |- ((dom` T):dom (dom` T)-->dom (id` T) -> (((id` T)` a) e. dom (dom` T) -> (D` ((id` T)` a)) e. ran (dom` T)))
26	1, 18, 25	3syl 24	. . . . . . 7 |- (T e. Ded -> (((id` T)` a) e. dom (dom` T) -> (D` ((id` T)` a)) e. ran (dom` T)))
27	16, 26	syld 30	. . . . . 6 |- (T e. Ded -> (a e. O -> (D` ((id` T)` a)) e. ran (dom` T)))
28	27	imp 377	. . . . 5 |- ((T e. Ded /\ a e. O) -> (D` ((id` T)` a)) e. ran (dom` T))
29		eleq1 1957	. . . . . . . 8 |- ((D` ((id` T)` a)) = a -> ((D` ((id` T)` a)) e. ran (dom` T) <-> a e. ran (dom` T)))
30	3	rneqi 4187	. . . . . . . . 9 |- ran D = ran (dom` T)
31	30	eleq2i 1961	. . . . . . . 8 |- (a e. ran D <-> a e. ran (dom` T))
32	29, 31	syl6bbr 597	. . . . . . 7 |- ((D` ((id` T)` a)) = a -> ((D` ((id` T)` a)) e. ran (dom` T) <-> a e. ran D))
33	32	biimpd 170	. . . . . 6 |- ((D` ((id` T)` a)) = a -> ((D` ((id` T)` a)) e. ran (dom` T) -> a e. ran D))
34	33	adantr 425	. . . . 5 |- (((D` ((id` T)` a)) = a /\ ((cod` T)` ((id` T)` a)) = a) -> ((D` ((id` T)` a)) e. ran (dom` T) -> a e. ran D))
35	10, 28, 34	sylc 83	. . . 4 |- ((T e. Ded /\ a e. O) -> a e. ran D)
36	35	ex 402	. . 3 |- (T e. Ded -> (a e. O -> a e. ran D))
37	36	ssrdv 2622	. 2 |- (T e. Ded -> O C_ ran D)
38	8, 37	eqssd 2633	1 |- (T e. Ded -> ran D = O)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   C_ wss 2593  dom cdm 3986  ran crn 3987  Fun wfun 3992  -->wf 3994  ` cfv 3998   Alg calg 15058  domcdom_ 15059  codccod_ 15060  idcid_ 15061   Ded cded 15081

This theorem is referenced by:  aidm2 15097

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-int 3215  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-opr 4886  df-1st 5020  df-2nd 5021  df-alg 15063  df-doma 15064  df-coda 15065  df-ida 15066  df-cmpa 15067  df-ded 15082

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