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Theorem coeq0 25997
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 5080 and coundir 5081 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 5077 . . 3  |-  Rel  ( A  o.  B )
2 relrn0 4844 . . 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 5085 . . 3  |-  ran  (  A  o.  B )  =  ran  (  A  |`  ran  B )
54eqeq1i 2260 . 2  |-  ( ran  (  A  o.  B
)  =  (/)  <->  ran  (  A  |`  ran  B )  =  (/) )
6 relres 4890 . . . 4  |-  Rel  ( A  |`  ran  B )
7 reldm0 4803 . . . 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 4844 . . . 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 4883 . . . . 5  |-  dom  (  A  |`  ran  B )  =  ( ran  B  i^i  dom  A )
12 incom 3269 . . . . 5  |-  ( ran 
B  i^i  dom  A )  =  ( dom  A  i^i  ran  B )
1311, 12eqtri 2273 . . . 4  |-  dom  (  A  |`  ran  B )  =  ( dom  A  i^i  ran  B )
1413eqeq1i 2260 . . 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 3077   (/)c0 3362   dom cdm 4580   ran crn 4581    |` cres 4582    o. ccom 4584   Rel wrel 4585
This theorem is referenced by:  coeq0i  25998  diophrw  26004
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600
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