Theorem coeq0 5521

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 5513 and coundir 5514 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 5510	. . 3
2		relrn0 5265	. . 3
3	1, 2	ax-mp 5	. 2
4		rnco 5518	. . 3
5	4	eqeq1i 2464	. 2
6		relres 5306	. . . 4
7		reldm0 5225	. . . 4
8	6, 7	ax-mp 5	. . 3
9		relrn0 5265	. . . 4
10	6, 9	ax-mp 5	. . 3
11		dmres 5299	. . . . 5
12		incom 3690	. . . . 5
13	11, 12	eqtri 2486	. . . 4
14	13	eqeq1i 2464	. . 3
15	8, 10, 14	3bitr3i 275	. 2
16	3, 5, 15	3bitri 271	1

Syntax hints:  <->wb 184  =wceq 1395  i^icin 3474  c0 3784  domcdm 5004  rancrn 5005  |`cres 5006  o.ccom 5008  Relwrel 5009

This theorem is referenced by:  coeq0i  30686  diophrw  30692  coemptyd  37760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016