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Theorem coeq0 26198
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 5161 and coundir 5162 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 5158 . . 3  |-  Rel  ( A  o.  B )
2 relrn0 4925 . . 3  |-  ( Rel  ( A  o.  B
)  ->  ( ( A  o.  B )  =  (/)  <->  ran  (  A  o.  B )  =  (/) ) )
31, 2ax-mp 10 . 2  |-  ( ( A  o.  B )  =  (/)  <->  ran  (  A  o.  B )  =  (/) )
4 rnco 5166 . . 3  |-  ran  (  A  o.  B )  =  ran  (  A  |`  ran  B )
54eqeq1i 2265 . 2  |-  ( ran  (  A  o.  B
)  =  (/)  <->  ran  (  A  |`  ran  B )  =  (/) )
6 relres 4971 . . . 4  |-  Rel  ( A  |`  ran  B )
7 reldm0 4884 . . . 4  |-  ( Rel  ( A  |`  ran  B
)  ->  ( ( A  |`  ran  B )  =  (/)  <->  dom  (  A  |`  ran  B )  =  (/) ) )
86, 7ax-mp 10 . . 3  |-  ( ( A  |`  ran  B )  =  (/)  <->  dom  (  A  |`  ran  B )  =  (/) )
9 relrn0 4925 . . . 4  |-  ( Rel  ( A  |`  ran  B
)  ->  ( ( A  |`  ran  B )  =  (/)  <->  ran  (  A  |`  ran  B )  =  (/) ) )
106, 9ax-mp 10 . . 3  |-  ( ( A  |`  ran  B )  =  (/)  <->  ran  (  A  |`  ran  B )  =  (/) )
11 dmres 4964 . . . . 5  |-  dom  (  A  |`  ran  B )  =  ( ran  B  i^i  dom  A )
12 incom 3336 . . . . 5  |-  ( ran 
B  i^i  dom  A )  =  ( dom  A  i^i  ran  B )
1311, 12eqtri 2278 . . . 4  |-  dom  (  A  |`  ran  B )  =  ( dom  A  i^i  ran  B )
1413eqeq1i 2265 . . 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 1619    i^i cin 3126   (/)c0 3430   dom cdm 4661   ran crn 4662    |` cres 4663    o. ccom 4665   Rel wrel 4666
This theorem is referenced by:  coeq0i  26199  diophrw  26205
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-br 3998  df-opab 4052  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681
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