Theorem coeq0 5644

Description: A composition of two relations is empty iff there is no overlap between the range of the second and the domain of the first. Useful in combination with coundi 5636 and coundir 5637 to prune meaningless terms in the result. (Contributed by Stefan O'Rear, 8-Oct-2014.)

Assertion

Ref	Expression
coeq0	|- ( ( A o. B ) = (/) <-> ( dom A i^i ran B ) = (/) )

Proof of Theorem coeq0

Step	Hyp	Ref	Expression
1		relco 5633	. . 3 |- Rel ( A o. B )
2		relrn0 5383	. . 3 |- ( Rel ( A o. B ) -> ( ( A o. B ) = (/) <-> ran ( A o. B ) = (/) ) )
3	1, 2	ax-mp 5	. 2 |- ( ( A o. B ) = (/) <-> ran ( A o. B ) = (/) )
4		rnco 5641	. . 3 |- ran ( A o. B ) = ran ( A |` ran B )
5	4	eqeq1i 2627	. 2 |- ( ran ( A o. B ) = (/) <-> ran ( A |` ran B ) = (/) )
6		relres 5426	. . . 4 |- Rel ( A |` ran B )
7		reldm0 5343	. . . 4 |- ( Rel ( A |` ran B ) -> ( ( A |` ran B ) = (/) <-> dom ( A |` ran B ) = (/) ) )
8	6, 7	ax-mp 5	. . 3 |- ( ( A |` ran B ) = (/) <-> dom ( A |` ran B ) = (/) )
9		relrn0 5383	. . . 4 |- ( Rel ( A |` ran B ) -> ( ( A |` ran B ) = (/) <-> ran ( A |` ran B ) = (/) ) )
10	6, 9	ax-mp 5	. . 3 |- ( ( A |` ran B ) = (/) <-> ran ( A |` ran B ) = (/) )
11		dmres 5419	. . . . 5 |- dom ( A |` ran B ) = ( ran B i^i dom A )
12		incom 3805	. . . . 5 |- ( ran B i^i dom A ) = ( dom A i^i ran B )
13	11, 12	eqtri 2644	. . . 4 |- dom ( A |` ran B ) = ( dom A i^i ran B )
14	13	eqeq1i 2627	. . 3 |- ( dom ( A |` ran B ) = (/) <-> ( dom A i^i ran B ) = (/) )
15	8, 10, 14	3bitr3i 290	. 2 |- ( ran ( A |` ran B ) = (/) <-> ( dom A i^i ran B ) = (/) )
16	3, 5, 15	3bitri 286	1 |- ( ( A o. B ) = (/) <-> ( dom A i^i ran B ) = (/) )

Syntax hints:   <-> wb 196   = wceq 1483   i^i cin 3573   (/)c0 3915   dom cdm 5114   ran crn 5115   |` cres 5116   o. ccom 5118   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  coemptyd  13718  coeq0i  37316  diophrw  37322  relexpnul  37970