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Theorem f1co 5411
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 5226 . . 3  |-  ( F : B -1-1-> C  <->  ( F : B --> C  /\  Fun  `' F ) )
2 df-f1 5226 . . 3  |-  ( G : A -1-1-> B  <->  ( G : A --> B  /\  Fun  `' G ) )
3 fco 5363 . . . . 5  |-  ( ( F : B --> C  /\  G : A --> B )  ->  ( F  o.  G ) : A --> C )
4 funco 5257 . . . . . . 7  |-  ( ( Fun  `' G  /\  Fun  `' F )  ->  Fun  ( `' G  o.  `' F ) )
5 cnvco 4864 . . . . . . . 8  |-  `' ( F  o.  G )  =  ( `' G  o.  `' F )
65funeqi 5241 . . . . . . 7  |-  ( Fun  `' ( F  o.  G )  <->  Fun  ( `' G  o.  `' F
) )
74, 6sylibr 205 . . . . . 6  |-  ( ( Fun  `' G  /\  Fun  `' F )  ->  Fun  `' ( F  o.  G
) )
87ancoms 441 . . . . 5  |-  ( ( Fun  `' F  /\  Fun  `' G )  ->  Fun  `' ( F  o.  G
) )
93, 8anim12i 551 . . . 4  |-  ( ( ( F : B --> C  /\  G : A --> B )  /\  ( Fun  `' F  /\  Fun  `' G ) )  -> 
( ( F  o.  G ) : A --> C  /\  Fun  `' ( F  o.  G ) ) )
109an4s 802 . . 3  |-  ( ( ( F : B --> C  /\  Fun  `' F
)  /\  ( G : A --> B  /\  Fun  `' G ) )  -> 
( ( F  o.  G ) : A --> C  /\  Fun  `' ( F  o.  G ) ) )
111, 2, 10syl2anb 467 . 2  |-  ( ( F : B -1-1-> C  /\  G : A -1-1-> B
)  ->  ( ( F  o.  G ) : A --> C  /\  Fun  `' ( F  o.  G
) ) )
12 df-f1 5226 . 2  |-  ( ( F  o.  G ) : A -1-1-> C  <->  ( ( F  o.  G ) : A --> C  /\  Fun  `' ( F  o.  G
) ) )
1311, 12sylibr 205 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 6    /\ wa 360   `'ccnv 4687    o. ccom 4692   Fun wfun 5215   -->wf 5217   -1-1->wf1 5218
This theorem is referenced by:  f1oco  5461  tposf12  6220  domtr  6909  dfac12lem2  7765  fin23lem28  7961  pwfseqlem5  8280  cofth  13803  gsumzf1o  15190  erdsze2lem2  23139
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  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 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  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-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226
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