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Theorem coeq0 26812
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 5373 and coundir 5374 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 5370 . . 3  |-  Rel  ( A  o.  B )
2 relrn0 5130 . . 3  |-  ( Rel  ( A  o.  B
)  ->  ( ( A  o.  B )  =  (/)  <->  ran  ( A  o.  B )  =  (/) ) )
31, 2ax-mp 8 . 2  |-  ( ( A  o.  B )  =  (/)  <->  ran  ( A  o.  B )  =  (/) )
4 rnco 5378 . . 3  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
54eqeq1i 2445 . 2  |-  ( ran  ( A  o.  B
)  =  (/)  <->  ran  ( A  |`  ran  B )  =  (/) )
6 relres 5176 . . . 4  |-  Rel  ( A  |`  ran  B )
7 reldm0 5089 . . . 4  |-  ( Rel  ( A  |`  ran  B
)  ->  ( ( A  |`  ran  B )  =  (/)  <->  dom  ( A  |`  ran  B )  =  (/) ) )
86, 7ax-mp 8 . . 3  |-  ( ( A  |`  ran  B )  =  (/)  <->  dom  ( A  |`  ran  B )  =  (/) )
9 relrn0 5130 . . . 4  |-  ( Rel  ( A  |`  ran  B
)  ->  ( ( A  |`  ran  B )  =  (/)  <->  ran  ( A  |`  ran  B )  =  (/) ) )
106, 9ax-mp 8 . . 3  |-  ( ( A  |`  ran  B )  =  (/)  <->  ran  ( A  |`  ran  B )  =  (/) )
11 dmres 5169 . . . . 5  |-  dom  ( A  |`  ran  B )  =  ( ran  B  i^i  dom  A )
12 incom 3535 . . . . 5  |-  ( ran 
B  i^i  dom  A )  =  ( dom  A  i^i  ran  B )
1311, 12eqtri 2458 . . . 4  |-  dom  ( A  |`  ran  B )  =  ( dom  A  i^i  ran  B )
1413eqeq1i 2445 . . 3  |-  ( dom  ( A  |`  ran  B
)  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
158, 10, 143bitr3i 268 . 2  |-  ( ran  ( A  |`  ran  B
)  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
163, 5, 153bitri 264 1  |-  ( ( A  o.  B )  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    i^i cin 3321   (/)c0 3630   dom cdm 4880   ran crn 4881    |` cres 4882    o. ccom 4884   Rel wrel 4885
This theorem is referenced by:  coeq0i  26813  diophrw  26819
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
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 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892
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