Theorem cmprelid2 14447

Description: Composition of a relation with a restriction of the identity to the field of the relation.

Assertion

Ref	Expression
cmprelid2	|- (Rel R -> (R o. ( _I |` U.U.R)) = R)

Proof of Theorem cmprelid2

Step	Hyp	Ref	Expression
1		relfld 4419	. . 3 |- (Rel R -> U.U.R = (dom R u. ran R))
2		reseq2 4219	. . . . . 6 |- (U.U.R = (dom R u. ran R) -> ( _I |` U.U.R) = ( _I |` (dom R u. ran R)))
3	2	coeq2d 4128	. . . . 5 |- (U.U.R = (dom R u. ran R) -> (R o. ( _I |` U.U.R)) = (R o. ( _I |` (dom R u. ran R))))
4	3	eqeq1d 1892	. . . 4 |- (U.U.R = (dom R u. ran R) -> ((R o. ( _I |` U.U.R)) = (R o. _I ) <-> (R o. ( _I |` (dom R u. ran R))) = (R o. _I )))
5		resundi 4229	. . . . 5 |- ( _I |` (dom R u. ran R)) = (( _I |` dom R) u. ( _I |` ran R))
6		coeq2 4124	. . . . . 6 |- (( _I |` (dom R u. ran R)) = (( _I |` dom R) u. ( _I |` ran R)) -> (R o. ( _I |` (dom R u. ran R))) = (R o. (( _I |` dom R) u. ( _I |` ran R))))
7		eqeq1 1890	. . . . . . 7 |- ((R o. ( _I |` (dom R u. ran R))) = (R o. (( _I |` dom R) u. ( _I |` ran R))) -> ((R o. ( _I |` (dom R u. ran R))) = (R o. _I ) <-> (R o. (( _I |` dom R) u. ( _I |` ran R))) = (R o. _I )))
8		resco 4402	. . . . . . . . 9 |- ((R o. _I ) |` dom R) = (R o. ( _I |` dom R))
9		coi1 4413	. . . . . . . . . 10 |- (Rel R -> (R o. _I ) = R)
10		resdm 4249	. . . . . . . . . . . 12 |- (Rel R -> (R |` dom R) = R)
11		eqtr 1904	. . . . . . . . . . . . . . 15 |- ((((R o. _I ) |` dom R) = (R |` dom R) /\ (R |` dom R) = R) -> ((R o. _I ) |` dom R) = R)
12		eqtr 1904	. . . . . . . . . . . . . . . . . . 19 |- (((R o. ( _I |` dom R)) = ((R o. _I ) |` dom R) /\ ((R o. _I ) |` dom R) = R) -> (R o. ( _I |` dom R)) = R)
13		uneq1 2748	. . . . . . . . . . . . . . . . . . . 20 |- ((R o. ( _I |` dom R)) = R -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R o. ( _I |` ran R))))
14		resco 4402	. . . . . . . . . . . . . . . . . . . . 21 |- ((R o. _I ) |` ran R) = (R o. ( _I |` ran R))
15		reseq1 4218	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((R o. _I ) = R -> ((R o. _I ) |` ran R) = (R |` ran R))
16		eqtr 1904	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (((R o. ( _I |` ran R)) = ((R o. _I ) |` ran R) /\ ((R o. _I ) |` ran R) = (R |` ran R)) -> (R o. ( _I |` ran R)) = (R |` ran R))
17		uneq2 2749	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ((R o. ( _I |` ran R)) = (R |` ran R) -> (R u. (R o. ( _I |` ran R))) = (R u. (R |` ran R)))
18		eqtr 1904	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ((((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R o. ( _I |` ran R))) /\ (R u. (R o. ( _I |` ran R))) = (R u. (R |` ran R))) -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R |` ran R)))
19		eqeq1 1890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R |` ran R)) -> (((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I ) <-> (R u. (R |` ran R)) = (R o. _I )))
20		resss 4237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- (R |` ran R) C_ R
21		ssequn2 2779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ((R |` ran R) C_ R <-> (R u. (R |` ran R)) = R)
22	20, 21	mpbi 206	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (R u. (R |` ran R)) = R
23	9, 22	syl6reqr 1947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- (Rel R -> (R u. (R |` ran R)) = (R o. _I ))
24	19, 23	syl5bir 227	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- (((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R |` ran R)) -> (Rel R -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I )))
25	18, 24	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ((((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R o. ( _I |` ran R))) /\ (R u. (R o. ( _I |` ran R))) = (R u. (R |` ran R))) -> (Rel R -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I )))
26	25	ex 402	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R o. ( _I |` ran R))) -> ((R u. (R o. ( _I |` ran R))) = (R u. (R |` ran R)) -> (Rel R -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I ))))
27	26	com3l 38	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ((R u. (R o. ( _I |` ran R))) = (R u. (R |` ran R)) -> (Rel R -> (((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R o. ( _I |` ran R))) -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I ))))
28	17, 27	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((R o. ( _I |` ran R)) = (R |` ran R) -> (Rel R -> (((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R o. ( _I |` ran R))) -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I ))))
29	16, 28	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (((R o. ( _I |` ran R)) = ((R o. _I ) |` ran R) /\ ((R o. _I ) |` ran R) = (R |` ran R)) -> (Rel R -> (((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R o. ( _I |` ran R))) -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I ))))
30	29	ex 402	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((R o. ( _I |` ran R)) = ((R o. _I ) |` ran R) -> (((R o. _I ) |` ran R) = (R |` ran R) -> (Rel R -> (((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R o. ( _I |` ran R))) -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I )))))
31	30	eqcoms 1887	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((R o. _I ) |` ran R) = (R o. ( _I |` ran R)) -> (((R o. _I ) |` ran R) = (R |` ran R) -> (Rel R -> (((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R o. ( _I |` ran R))) -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I )))))
32	31	com3l 38	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((R o. _I ) |` ran R) = (R |` ran R) -> (Rel R -> (((R o. _I ) |` ran R) = (R o. ( _I |` ran R)) -> (((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R o. ( _I |` ran R))) -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I )))))
33	15, 32	syl 12	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((R o. _I ) = R -> (Rel R -> (((R o. _I ) |` ran R) = (R o. ( _I |` ran R)) -> (((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R o. ( _I |` ran R))) -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I )))))
34	9, 33	mpcom 60	. . . . . . . . . . . . . . . . . . . . . 22 |- (Rel R -> (((R o. _I ) |` ran R) = (R o. ( _I |` ran R)) -> (((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R o. ( _I |` ran R))) -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I ))))
35	34	com3l 38	. . . . . . . . . . . . . . . . . . . . 21 |- (((R o. _I ) |` ran R) = (R o. ( _I |` ran R)) -> (((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R o. ( _I |` ran R))) -> (Rel R -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I ))))
36	14, 35	ax-mp 7	. . . . . . . . . . . . . . . . . . . 20 |- (((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R u. (R o. ( _I |` ran R))) -> (Rel R -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I )))
37	13, 36	syl 12	. . . . . . . . . . . . . . . . . . 19 |- ((R o. ( _I |` dom R)) = R -> (Rel R -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I )))
38	12, 37	syl 12	. . . . . . . . . . . . . . . . . 18 |- (((R o. ( _I |` dom R)) = ((R o. _I ) |` dom R) /\ ((R o. _I ) |` dom R) = R) -> (Rel R -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I )))
39	38	ex 402	. . . . . . . . . . . . . . . . 17 |- ((R o. ( _I |` dom R)) = ((R o. _I ) |` dom R) -> (((R o. _I ) |` dom R) = R -> (Rel R -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I ))))
40	39	eqcoms 1887	. . . . . . . . . . . . . . . 16 |- (((R o. _I ) |` dom R) = (R o. ( _I |` dom R)) -> (((R o. _I ) |` dom R) = R -> (Rel R -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I ))))
41	40	com3l 38	. . . . . . . . . . . . . . 15 |- (((R o. _I ) |` dom R) = R -> (Rel R -> (((R o. _I ) |` dom R) = (R o. ( _I |` dom R)) -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I ))))
42	11, 41	syl 12	. . . . . . . . . . . . . 14 |- ((((R o. _I ) |` dom R) = (R |` dom R) /\ (R |` dom R) = R) -> (Rel R -> (((R o. _I ) |` dom R) = (R o. ( _I |` dom R)) -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I ))))
43	42	ex 402	. . . . . . . . . . . . 13 |- (((R o. _I ) |` dom R) = (R |` dom R) -> ((R |` dom R) = R -> (Rel R -> (((R o. _I ) |` dom R) = (R o. ( _I |` dom R)) -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I )))))
44	43	com3l 38	. . . . . . . . . . . 12 |- ((R |` dom R) = R -> (Rel R -> (((R o. _I ) |` dom R) = (R |` dom R) -> (((R o. _I ) |` dom R) = (R o. ( _I |` dom R)) -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I )))))
45	10, 44	mpcom 60	. . . . . . . . . . 11 |- (Rel R -> (((R o. _I ) |` dom R) = (R |` dom R) -> (((R o. _I ) |` dom R) = (R o. ( _I |` dom R)) -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I ))))
46		reseq1 4218	. . . . . . . . . . 11 |- ((R o. _I ) = R -> ((R o. _I ) |` dom R) = (R |` dom R))
47	45, 46	syl5com 63	. . . . . . . . . 10 |- ((R o. _I ) = R -> (Rel R -> (((R o. _I ) |` dom R) = (R o. ( _I |` dom R)) -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I ))))
48	9, 47	mpcom 60	. . . . . . . . 9 |- (Rel R -> (((R o. _I ) |` dom R) = (R o. ( _I |` dom R)) -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I )))
49	8, 48	mpi 55	. . . . . . . 8 |- (Rel R -> ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R))) = (R o. _I ))
50		coundi 4396	. . . . . . . 8 |- (R o. (( _I |` dom R) u. ( _I |` ran R))) = ((R o. ( _I |` dom R)) u. (R o. ( _I |` ran R)))
51	49, 50	syl5eq 1940	. . . . . . 7 |- (Rel R -> (R o. (( _I |` dom R) u. ( _I |` ran R))) = (R o. _I ))
52	7, 51	syl5bir 227	. . . . . 6 |- ((R o. ( _I |` (dom R u. ran R))) = (R o. (( _I |` dom R) u. ( _I |` ran R))) -> (Rel R -> (R o. ( _I |` (dom R u. ran R))) = (R o. _I )))
53	6, 52	syl 12	. . . . 5 |- (( _I |` (dom R u. ran R)) = (( _I |` dom R) u. ( _I |` ran R)) -> (Rel R -> (R o. ( _I |` (dom R u. ran R))) = (R o. _I )))
54	5, 53	ax-mp 7	. . . 4 |- (Rel R -> (R o. ( _I |` (dom R u. ran R))) = (R o. _I ))
55	4, 54	syl5bir 227	. . 3 |- (U.U.R = (dom R u. ran R) -> (Rel R -> (R o. ( _I |` U.U.R)) = (R o. _I )))
56	1, 55	mpcom 60	. 2 |- (Rel R -> (R o. ( _I |` U.U.R)) = (R o. _I ))
57	56, 9	eqtrd 1925	1 |- (Rel R -> (R o. ( _I |` U.U.R)) = R)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   u. cun 2591   C_ wss 2593  U.cuni 3177   _I cid 3582  dom cdm 3986  ran crn 3987   |` cres 3988   o. ccom 3990  Rel wrel 3991

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-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

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