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Theorem coeq0 5351
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 5343 and coundir 5344 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 5340 . . 3  |-  Rel  ( A  o.  B )
2 relrn0 5098 . . 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 5348 . . 3  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
54eqeq1i 2476 . 2  |-  ( ran  ( A  o.  B
)  =  (/)  <->  ran  ( A  |`  ran  B )  =  (/) )
6 relres 5138 . . . 4  |-  Rel  ( A  |`  ran  B )
7 reldm0 5058 . . . 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 5098 . . . 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 5131 . . . . 5  |-  dom  ( A  |`  ran  B )  =  ( ran  B  i^i  dom  A )
12 incom 3616 . . . . 5  |-  ( ran 
B  i^i  dom  A )  =  ( dom  A  i^i  ran  B )
1311, 12eqtri 2493 . . . 4  |-  dom  ( A  |`  ran  B )  =  ( dom  A  i^i  ran  B )
1413eqeq1i 2476 . . 3  |-  ( dom  ( A  |`  ran  B
)  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
158, 10, 143bitr3i 283 . 2  |-  ( ran  ( A  |`  ran  B
)  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
163, 5, 153bitri 279 1  |-  ( ( A  o.  B )  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    = wceq 1452    i^i cin 3389   (/)c0 3722   dom cdm 4839   ran crn 4840    |` cres 4841    o. ccom 4843   Rel wrel 4844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851
This theorem is referenced by:  coemptyd  13118  coeq0i  35666  diophrw  35672  relexpnul  36341
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