Theorem rdmob 10652

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 10647	. . 3 |- (T e. Ded -> T e. Alg)
2		eqid 1478	. . . 4 |- dom D = dom D
3		rdmob.2	. . . 4 |- D = (dom` T)
4		rdmob.1	. . . 4 |- O = dom (id` T)
5		eqid 1478	. . . 4 |- (id` T) = (id` T)
6	2, 3, 4, 5	doma 10632	. . 3 |- (T e. Alg -> D:dom D-->O)
7		frn 3639	. . 3 |- (D:dom D-->O -> ran D (_ O)
8	1, 6, 7	3syl 20	. 2 |- (T e. Ded -> ran D (_ O)
9		eleq1 1537	. . . . . . . 8 |- ((D` ((id` T)` a)) = a -> ((D` ((id` T)` a)) e. ran (dom` T) <-> a e. ran (dom` T)))
10	3	rneqi 3346	. . . . . . . . 9 |- ran D = ran (dom` T)
11	10	eleq2i 1541	. . . . . . . 8 |- (a e. ran D <-> a e. ran (dom` T))
12	9, 11	syl6bbr 540	. . . . . . 7 |- ((D` ((id` T)` a)) = a -> ((D` ((id` T)` a)) e. ran (dom` T) <-> a e. ran D))
13	12	biimpd 153	. . . . . 6 |- ((D` ((id` T)` a)) = a -> ((D` ((id` T)` a)) e. ran (dom` T) -> a e. ran D))
14	13	adantr 391	. . . . 5 |- (((D` ((id` T)` a)) = a /\ ((cod` T)` ((id` T)` a)) = a) -> ((D` ((id` T)` a)) e. ran (dom` T) -> a e. ran D))
15		eqid 1478	. . . . . 6 |- (cod` T) = (cod` T)
16	4, 3, 5, 15	idosd 10648	. . . . 5 |- ((T e. Ded /\ a e. O) -> ((D` ((id` T)` a)) = a /\ ((cod` T)` ((id` T)` a)) = a))
17		eqid 1478	. . . . . . . . 9 |- dom (dom` T) = dom (dom` T)
18		eqid 1478	. . . . . . . . 9 |- (dom` T) = (dom` T)
19	17, 18, 4, 5	ida 10634	. . . . . . . 8 |- (T e. Alg -> (id` T):O-->dom (dom` T))
20		ffvelrn 3820	. . . . . . . . 9 |- (((id` T):O-->dom (dom` T) /\ a e. O) -> ((id` T)` a) e. dom (dom` T))
21	20	ex 373	. . . . . . . 8 |- ((id` T):O-->dom (dom` T) -> (a e. O -> ((id` T)` a) e. dom (dom` T)))
22	1, 19, 21	3syl 20	. . . . . . 7 |- (T e. Ded -> (a e. O -> ((id` T)` a) e. dom (dom` T)))
23		eqid 1478	. . . . . . . . 9 |- dom (id` T) = dom (id` T)
24	17, 18, 23, 5	doma 10632	. . . . . . . 8 |- (T e. Alg -> (dom` T):dom (dom` T)-->dom (id` T))
25		ffun 3635	. . . . . . . . . 10 |- ((dom` T):dom (dom` T)-->dom (id` T) -> Fun (dom` T))
26		fvelrn 3818	. . . . . . . . . . 11 |- ((Fun (dom` T) /\ ((id` T)` a) e. dom (dom` T)) -> ((dom` T)` ((id` T)` a)) e. ran (dom` T))
27	26	ex 373	. . . . . . . . . 10 |- (Fun (dom` T) -> (((id` T)` a) e. dom (dom` T) -> ((dom` T)` ((id` T)` a)) e. ran (dom` T)))
28	25, 27	syl 10	. . . . . . . . 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)))
29	3	fveq1i 3731	. . . . . . . . . 10 |- (D` ((id` T)` a)) = ((dom` T)` ((id` T)` a))
30	29	eleq1i 1540	. . . . . . . . 9 |- ((D` ((id` T)` a)) e. ran (dom` T) <-> ((dom` T)` ((id` T)` a)) e. ran (dom` T))
31	28, 30	syl6ibr 213	. . . . . . . 8 |- ((dom` T):dom (dom` T)-->dom (id` T) -> (((id` T)` a) e. dom (dom` T) -> (D` ((id` T)` a)) e. ran (dom` T)))
32	1, 24, 31	3syl 20	. . . . . . 7 |- (T e. Ded -> (((id` T)` a) e. dom (dom` T) -> (D` ((id` T)` a)) e. ran (dom` T)))
33	22, 32	syld 27	. . . . . 6 |- (T e. Ded -> (a e. O -> (D` ((id` T)` a)) e. ran (dom` T)))
34	33	imp 350	. . . . 5 |- ((T e. Ded /\ a e. O) -> (D` ((id` T)` a)) e. ran (dom` T))
35	14, 16, 34	sylc 68	. . . 4 |- ((T e. Ded /\ a e. O) -> a e. ran D)
36	35	ex 373	. . 3 |- (T e. Ded -> (a e. O -> a e. ran D))
37	36	ssrdv 2073	. 2 |- (T e. Ded -> O (_ ran D)
38	8, 37	eqssd 2082	1 |- (T e. Ded -> ran D = O)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960   (_ wss 2050  dom cdm 3176  ran crn 3177  Fun wfun 3182  -->wf 3184  ` cfv 3188  Algcalg 10614  domcdom_ 10615  codccod_ 10616  idcid_ 10617  Dedcded 10638

This theorem is referenced by:  aidm 10654

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  ax-un 2872

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-rex 1653  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-uni 2508  df-int 2538  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-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-opr 3971  df-1st 4085  df-2nd 4086  df-alg 10619  df-doma 10620  df-coda 10621  df-ida 10622  df-cmpa 10623  df-ded 10639

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