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Theorem coeq0 5363
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 5355 and coundir 5356 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
StepHypRef Expression
1 relco 5352 . . 3  |-  Rel  ( A  o.  B )
2 relrn0 5111 . . 3  |-  ( Rel  ( A  o.  B
)  ->  ( ( A  o.  B )  =  (/)  <->  ran  ( A  o.  B )  =  (/) ) )
31, 2ax-mp 5 . 2  |-  ( ( A  o.  B )  =  (/)  <->  ran  ( A  o.  B )  =  (/) )
4 rnco 5360 . . 3  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
54eqeq1i 2429 . 2  |-  ( ran  ( A  o.  B
)  =  (/)  <->  ran  ( A  |`  ran  B )  =  (/) )
6 relres 5151 . . . 4  |-  Rel  ( A  |`  ran  B )
7 reldm0 5071 . . . 4  |-  ( Rel  ( A  |`  ran  B
)  ->  ( ( A  |`  ran  B )  =  (/)  <->  dom  ( A  |`  ran  B )  =  (/) ) )
86, 7ax-mp 5 . . 3  |-  ( ( A  |`  ran  B )  =  (/)  <->  dom  ( A  |`  ran  B )  =  (/) )
9 relrn0 5111 . . . 4  |-  ( Rel  ( A  |`  ran  B
)  ->  ( ( A  |`  ran  B )  =  (/)  <->  ran  ( A  |`  ran  B )  =  (/) ) )
106, 9ax-mp 5 . . 3  |-  ( ( A  |`  ran  B )  =  (/)  <->  ran  ( A  |`  ran  B )  =  (/) )
11 dmres 5144 . . . . 5  |-  dom  ( A  |`  ran  B )  =  ( ran  B  i^i  dom  A )
12 incom 3655 . . . . 5  |-  ( ran 
B  i^i  dom  A )  =  ( dom  A  i^i  ran  B )
1311, 12eqtri 2451 . . . 4  |-  dom  ( A  |`  ran  B )  =  ( dom  A  i^i  ran  B )
1413eqeq1i 2429 . . 3  |-  ( dom  ( A  |`  ran  B
)  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
158, 10, 143bitr3i 278 . 2  |-  ( ran  ( A  |`  ran  B
)  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
163, 5, 153bitri 274 1  |-  ( ( A  o.  B )  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    = wceq 1437    i^i cin 3435   (/)c0 3761   dom cdm 4853   ran crn 4854    |` cres 4855    o. ccom 4857   Rel wrel 4858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865
This theorem is referenced by:  coemptyd  13043  coeq0i  35564  diophrw  35570  relexpnul  36240
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