Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  coeq0 Unicode version

Theorem coeq0 26230
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 5172 and coundir 5173 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 5169 . . 3  |-  Rel  ( A  o.  B )
2 relrn0 4936 . . 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 5177 . . 3  |-  ran  (  A  o.  B )  =  ran  (  A  |`  ran  B )
54eqeq1i 2291 . 2  |-  ( ran  (  A  o.  B
)  =  (/)  <->  ran  (  A  |`  ran  B )  =  (/) )
6 relres 4982 . . . 4  |-  Rel  ( A  |`  ran  B )
7 reldm0 4895 . . . 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 4936 . . . 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 4975 . . . . 5  |-  dom  (  A  |`  ran  B )  =  ( ran  B  i^i  dom  A )
12 incom 3362 . . . . 5  |-  ( ran 
B  i^i  dom  A )  =  ( dom  A  i^i  ran  B )
1311, 12eqtri 2304 . . . 4  |-  dom  (  A  |`  ran  B )  =  ( dom  A  i^i  ran  B )
1413eqeq1i 2291 . . 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 1624    i^i cin 3152   (/)c0 3456   dom cdm 4688   ran crn 4689    |` cres 4690    o. ccom 4692   Rel wrel 4693
This theorem is referenced by:  coeq0i  26231  diophrw  26237
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700
  Copyright terms: Public domain W3C validator