Theorem coeq0 26801

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 5363 and coundir 5364 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 5360	. . 3
2		relrn0 5120	. . 3
3	1, 2	ax-mp 8	. 2
4		rnco 5368	. . 3
5	4	eqeq1i 2442	. 2
6		relres 5166	. . . 4
7		reldm0 5079	. . . 4
8	6, 7	ax-mp 8	. . 3
9		relrn0 5120	. . . 4
10	6, 9	ax-mp 8	. . 3
11		dmres 5159	. . . . 5
12		incom 3525	. . . . 5
13	11, 12	eqtri 2455	. . . 4
14	13	eqeq1i 2442	. . 3
15	8, 10, 14	3bitr3i 267	. 2
16	3, 5, 15	3bitri 263	1

Syntax hints:  <->wb 177  =wceq 1652  i^icin 3311  c0 3620  domcdm 4870  rancrn 4871  |`cres 4872  o.ccom 4874  Relwrel 4875

This theorem is referenced by:  coeq0i  26802  diophrw  26808

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882