Theorem coeq0 27147

Description: A composition of two relations is empty iff there is no overlap betwen the range of the second and the domain of the first. Useful in combination with coundi 5421 and coundir 5422 to prune meaningless terms in the result. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Assertion

Ref	Expression
coeq0

Proof of Theorem coeq0

Step	Hyp	Ref	Expression
1		relco 5418	. . 3
2		relrn0 5175	. . 3
3	1, 2	ax-mp 5	. 2
4		rnco 5426	. . 3
5	4	eqeq1i 2454	. 2
6		relres 5222	. . . 4
7		reldm0 5135	. . . 4
8	6, 7	ax-mp 5	. . 3
9		relrn0 5175	. . . 4
10	6, 9	ax-mp 5	. . 3
11		dmres 5215	. . . . 5
12		incom 3526	. . . . 5
13	11, 12	eqtri 2467	. . . 4
14	13	eqeq1i 2454	. . 3
15	8, 10, 14	3bitr3i 268	. 2
16	3, 5, 15	3bitri 264	1

Syntax hints:  <->wb 178  =wceq 1654  i^icin 3312  c0 3620  domcdm 4923  rancrn 4924  |`cres 4925  o.ccom 4927  Relwrel 4928

This theorem is referenced by:  coeq0i  27148  diophrw  27154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1955  ax-ext 2428  ax-sep 4368  ax-nul 4376  ax-pr 4446

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2296  df-mo 2297  df-clab 2434  df-cleq 2440  df-clel 2443  df-nfc 2572  df-ne 2612  df-ral 2721  df-rex 2722  df-rab 2725  df-v 2971  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3621  df-if 3770  df-sn 3851  df-pr 3852  df-op 3854  df-br 4248  df-opab 4306  df-xp 4929  df-rel 4930  df-cnv 4931  df-co 4932  df-dm 4933  df-rn 4934  df-res 4935