Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  coeq0i Unicode version

Theorem coeq0i 26935
Description: coeq0 26934 but without explicitly introducing domain and range symbols. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Assertion
Ref Expression
coeq0i  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( A  o.  B )  =  (/) )

Proof of Theorem coeq0i
StepHypRef Expression
1 frn 5411 . . . . . 6  |-  ( B : E --> F  ->  ran  B  C_  F )
213ad2ant2 977 . . . . 5  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ran  B 
C_  F )
3 sslin 3408 . . . . 5  |-  ( ran 
B  C_  F  ->  ( dom  A  i^i  ran  B )  C_  ( dom  A  i^i  F ) )
42, 3syl 15 . . . 4  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  ran  B
)  C_  ( dom  A  i^i  F ) )
5 fdm 5409 . . . . . . 7  |-  ( A : C --> D  ->  dom  A  =  C )
653ad2ant1 976 . . . . . 6  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  dom  A  =  C )
76ineq1d 3382 . . . . 5  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  F )  =  ( C  i^i  F ) )
8 simp3 957 . . . . 5  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( C  i^i  F )  =  (/) )
97, 8eqtrd 2328 . . . 4  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  F )  =  (/) )
104, 9sseqtrd 3227 . . 3  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  ran  B
)  C_  (/) )
11 ss0 3498 . . 3  |-  ( ( dom  A  i^i  ran  B )  C_  (/)  ->  ( dom  A  i^i  ran  B
)  =  (/) )
1210, 11syl 15 . 2  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( dom  A  i^i  ran  B
)  =  (/) )
13 coeq0 26934 . 2  |-  ( ( A  o.  B )  =  (/)  <->  ( dom  A  i^i  ran  B )  =  (/) )
1412, 13sylibr 203 1  |-  ( ( A : C --> D  /\  B : E --> F  /\  ( C  i^i  F )  =  (/) )  ->  ( A  o.  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    i^i cin 3164    C_ wss 3165   (/)c0 3468   dom cdm 4705   ran crn 4706    o. ccom 4709   -->wf 5267
This theorem is referenced by:  diophren  26999
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  df-fn 5274  df-f 5275
  Copyright terms: Public domain W3C validator