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Theorem coeq0 26831
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 5174 and coundir 5175 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 5171 . . 3  |-  Rel  ( A  o.  B )
2 relrn0 4937 . . 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 5179 . . 3  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
54eqeq1i 2290 . 2  |-  ( ran  ( A  o.  B
)  =  (/)  <->  ran  ( A  |`  ran  B )  =  (/) )
6 relres 4983 . . . 4  |-  Rel  ( A  |`  ran  B )
7 reldm0 4896 . . . 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 4937 . . . 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 4976 . . . . 5  |-  dom  ( A  |`  ran  B )  =  ( ran  B  i^i  dom  A )
12 incom 3361 . . . . 5  |-  ( ran 
B  i^i  dom  A )  =  ( dom  A  i^i  ran  B )
1311, 12eqtri 2303 . . . 4  |-  dom  ( A  |`  ran  B )  =  ( dom  A  i^i  ran  B )
1413eqeq1i 2290 . . 3  |-  ( dom  ( A  |`  ran  B
)  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
158, 10, 143bitr3i 266 . 2  |-  ( ran  ( A  |`  ran  B
)  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
163, 5, 153bitri 262 1  |-  ( ( A  o.  B )  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    i^i cin 3151   (/)c0 3455   dom cdm 4689   ran crn 4690    |` cres 4691    o. ccom 4693   Rel wrel 4694
This theorem is referenced by:  coeq0i  26832  diophrw  26838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701
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