MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  foco Unicode version

Theorem foco 5477
Description: Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
Assertion
Ref Expression
foco  |-  ( ( F : B -onto-> C  /\  G : A -onto-> B
)  ->  ( F  o.  G ) : A -onto-> C )

Proof of Theorem foco
StepHypRef Expression
1 dffo2 5471 . . 3  |-  ( F : B -onto-> C  <->  ( F : B --> C  /\  ran  F  =  C ) )
2 dffo2 5471 . . 3  |-  ( G : A -onto-> B  <->  ( G : A --> B  /\  ran  G  =  B ) )
3 fco 5414 . . . . 5  |-  ( ( F : B --> C  /\  G : A --> B )  ->  ( F  o.  G ) : A --> C )
43ad2ant2r 727 . . . 4  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ( G : A --> B  /\  ran  G  =  B ) )  ->  ( F  o.  G ) : A --> C )
5 fdm 5409 . . . . . . . 8  |-  ( F : B --> C  ->  dom  F  =  B )
6 eqtr3 2315 . . . . . . . 8  |-  ( ( dom  F  =  B  /\  ran  G  =  B )  ->  dom  F  =  ran  G )
75, 6sylan 457 . . . . . . 7  |-  ( ( F : B --> C  /\  ran  G  =  B )  ->  dom  F  =  ran  G )
8 rncoeq 4964 . . . . . . . . 9  |-  ( dom 
F  =  ran  G  ->  ran  ( F  o.  G )  =  ran  F )
98eqeq1d 2304 . . . . . . . 8  |-  ( dom 
F  =  ran  G  ->  ( ran  ( F  o.  G )  =  C  <->  ran  F  =  C ) )
109biimpar 471 . . . . . . 7  |-  ( ( dom  F  =  ran  G  /\  ran  F  =  C )  ->  ran  ( F  o.  G
)  =  C )
117, 10sylan 457 . . . . . 6  |-  ( ( ( F : B --> C  /\  ran  G  =  B )  /\  ran  F  =  C )  ->  ran  ( F  o.  G
)  =  C )
1211an32s 779 . . . . 5  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ran  G  =  B )  ->  ran  ( F  o.  G
)  =  C )
1312adantrl 696 . . . 4  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ( G : A --> B  /\  ran  G  =  B ) )  ->  ran  ( F  o.  G )  =  C )
144, 13jca 518 . . 3  |-  ( ( ( F : B --> C  /\  ran  F  =  C )  /\  ( G : A --> B  /\  ran  G  =  B ) )  ->  ( ( F  o.  G ) : A --> C  /\  ran  ( F  o.  G
)  =  C ) )
151, 2, 14syl2anb 465 . 2  |-  ( ( F : B -onto-> C  /\  G : A -onto-> B
)  ->  ( ( F  o.  G ) : A --> C  /\  ran  ( F  o.  G
)  =  C ) )
16 dffo2 5471 . 2  |-  ( ( F  o.  G ) : A -onto-> C  <->  ( ( F  o.  G ) : A --> C  /\  ran  ( F  o.  G
)  =  C ) )
1715, 16sylibr 203 1  |-  ( ( F : B -onto-> C  /\  G : A -onto-> B
)  ->  ( F  o.  G ) : A -onto-> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632   dom cdm 4705   ran crn 4706    o. ccom 4709   -->wf 5267   -onto->wfo 5269
This theorem is referenced by:  f1oco  5512  wdomtr  7305  fin1a2lem7  8048  cofull  13824  uniiccdif  18949
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-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277
  Copyright terms: Public domain W3C validator