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Theorem f1co 5462
Description: Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.)
Assertion
Ref Expression
f1co  |-  ( ( F : B -1-1-> C  /\  G : A -1-1-> B
)  ->  ( F  o.  G ) : A -1-1-> C )

Proof of Theorem f1co
StepHypRef Expression
1 df-f1 5276 . . 3  |-  ( F : B -1-1-> C  <->  ( F : B --> C  /\  Fun  `' F ) )
2 df-f1 5276 . . 3  |-  ( G : A -1-1-> B  <->  ( G : A --> B  /\  Fun  `' G ) )
3 fco 5414 . . . . 5  |-  ( ( F : B --> C  /\  G : A --> B )  ->  ( F  o.  G ) : A --> C )
4 funco 5308 . . . . . . 7  |-  ( ( Fun  `' G  /\  Fun  `' F )  ->  Fun  ( `' G  o.  `' F ) )
5 cnvco 4881 . . . . . . . 8  |-  `' ( F  o.  G )  =  ( `' G  o.  `' F )
65funeqi 5291 . . . . . . 7  |-  ( Fun  `' ( F  o.  G )  <->  Fun  ( `' G  o.  `' F
) )
74, 6sylibr 203 . . . . . 6  |-  ( ( Fun  `' G  /\  Fun  `' F )  ->  Fun  `' ( F  o.  G
) )
87ancoms 439 . . . . 5  |-  ( ( Fun  `' F  /\  Fun  `' G )  ->  Fun  `' ( F  o.  G
) )
93, 8anim12i 549 . . . 4  |-  ( ( ( F : B --> C  /\  G : A --> B )  /\  ( Fun  `' F  /\  Fun  `' G ) )  -> 
( ( F  o.  G ) : A --> C  /\  Fun  `' ( F  o.  G ) ) )
109an4s 799 . . 3  |-  ( ( ( F : B --> C  /\  Fun  `' F
)  /\  ( G : A --> B  /\  Fun  `' G ) )  -> 
( ( F  o.  G ) : A --> C  /\  Fun  `' ( F  o.  G ) ) )
111, 2, 10syl2anb 465 . 2  |-  ( ( F : B -1-1-> C  /\  G : A -1-1-> B
)  ->  ( ( F  o.  G ) : A --> C  /\  Fun  `' ( F  o.  G
) ) )
12 df-f1 5276 . 2  |-  ( ( F  o.  G ) : A -1-1-> C  <->  ( ( F  o.  G ) : A --> C  /\  Fun  `' ( F  o.  G
) ) )
1311, 12sylibr 203 1  |-  ( ( F : B -1-1-> C  /\  G : A -1-1-> B
)  ->  ( F  o.  G ) : A -1-1-> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   `'ccnv 4704    o. ccom 4709   Fun wfun 5265   -->wf 5267   -1-1->wf1 5268
This theorem is referenced by:  f1oco  5512  tposf12  6275  domtr  6930  dfac12lem2  7786  fin23lem28  7982  pwfseqlem5  8301  cofth  13825  gsumzf1o  15212  erdsze2lem2  23750
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-f1 5276
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