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Theorem coeq0 26934
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 5190 and coundir 5191 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 5187 . . 3  |-  Rel  ( A  o.  B )
2 relrn0 4953 . . 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 5195 . . 3  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
54eqeq1i 2303 . 2  |-  ( ran  ( A  o.  B
)  =  (/)  <->  ran  ( A  |`  ran  B )  =  (/) )
6 relres 4999 . . . 4  |-  Rel  ( A  |`  ran  B )
7 reldm0 4912 . . . 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 4953 . . . 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 4992 . . . . 5  |-  dom  ( A  |`  ran  B )  =  ( ran  B  i^i  dom  A )
12 incom 3374 . . . . 5  |-  ( ran 
B  i^i  dom  A )  =  ( dom  A  i^i  ran  B )
1311, 12eqtri 2316 . . . 4  |-  dom  ( A  |`  ran  B )  =  ( dom  A  i^i  ran  B )
1413eqeq1i 2303 . . 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 1632    i^i cin 3164   (/)c0 3468   dom cdm 4705   ran crn 4706    |` cres 4707    o. ccom 4709   Rel wrel 4710
This theorem is referenced by:  coeq0i  26935  diophrw  26941
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717
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