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Theorem f1co 5639
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 5450 . . 3  |-  ( F : B -1-1-> C  <->  ( F : B --> C  /\  Fun  `' F ) )
2 df-f1 5450 . . 3  |-  ( G : A -1-1-> B  <->  ( G : A --> B  /\  Fun  `' G ) )
3 fco 5591 . . . . 5  |-  ( ( F : B --> C  /\  G : A --> B )  ->  ( F  o.  G ) : A --> C )
4 funco 5482 . . . . . . 7  |-  ( ( Fun  `' G  /\  Fun  `' F )  ->  Fun  ( `' G  o.  `' F ) )
5 cnvco 5047 . . . . . . . 8  |-  `' ( F  o.  G )  =  ( `' G  o.  `' F )
65funeqi 5465 . . . . . . 7  |-  ( Fun  `' ( F  o.  G )  <->  Fun  ( `' G  o.  `' F
) )
74, 6sylibr 204 . . . . . 6  |-  ( ( Fun  `' G  /\  Fun  `' F )  ->  Fun  `' ( F  o.  G
) )
87ancoms 440 . . . . 5  |-  ( ( Fun  `' F  /\  Fun  `' G )  ->  Fun  `' ( F  o.  G
) )
93, 8anim12i 550 . . . 4  |-  ( ( ( F : B --> C  /\  G : A --> B )  /\  ( Fun  `' F  /\  Fun  `' G ) )  -> 
( ( F  o.  G ) : A --> C  /\  Fun  `' ( F  o.  G ) ) )
109an4s 800 . . 3  |-  ( ( ( F : B --> C  /\  Fun  `' F
)  /\  ( G : A --> B  /\  Fun  `' G ) )  -> 
( ( F  o.  G ) : A --> C  /\  Fun  `' ( F  o.  G ) ) )
111, 2, 10syl2anb 466 . 2  |-  ( ( F : B -1-1-> C  /\  G : A -1-1-> B
)  ->  ( ( F  o.  G ) : A --> C  /\  Fun  `' ( F  o.  G
) ) )
12 df-f1 5450 . 2  |-  ( ( F  o.  G ) : A -1-1-> C  <->  ( ( F  o.  G ) : A --> C  /\  Fun  `' ( F  o.  G
) ) )
1311, 12sylibr 204 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 359   `'ccnv 4868    o. ccom 4873   Fun wfun 5439   -->wf 5441   -1-1->wf1 5442
This theorem is referenced by:  f1oco  5689  tposf12  6495  domtr  7151  dfac12lem2  8013  fin23lem28  8209  pwfseqlem5  8527  cofth  14120  gsumzf1o  15507  erdsze2lem2  24878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450
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