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Theorem 1stcof 6366
Description: Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
Assertion
Ref Expression
1stcof  |-  ( F : A --> ( B  X.  C )  -> 
( 1st  o.  F
) : A --> B )

Proof of Theorem 1stcof
StepHypRef Expression
1 fo1st 6358 . . . 4  |-  1st : _V -onto-> _V
2 fofn 5647 . . . 4  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 8 . . 3  |-  1st  Fn  _V
4 ffn 5583 . . . 4  |-  ( F : A --> ( B  X.  C )  ->  F  Fn  A )
5 dffn2 5584 . . . 4  |-  ( F  Fn  A  <->  F : A
--> _V )
64, 5sylib 189 . . 3  |-  ( F : A --> ( B  X.  C )  ->  F : A --> _V )
7 fnfco 5601 . . 3  |-  ( ( 1st  Fn  _V  /\  F : A --> _V )  ->  ( 1st  o.  F
)  Fn  A )
83, 6, 7sylancr 645 . 2  |-  ( F : A --> ( B  X.  C )  -> 
( 1st  o.  F
)  Fn  A )
9 rnco 5368 . . 3  |-  ran  ( 1st  o.  F )  =  ran  ( 1st  |`  ran  F
)
10 frn 5589 . . . . 5  |-  ( F : A --> ( B  X.  C )  ->  ran  F  C_  ( B  X.  C ) )
11 ssres2 5165 . . . . 5  |-  ( ran 
F  C_  ( B  X.  C )  ->  ( 1st  |`  ran  F ) 
C_  ( 1st  |`  ( B  X.  C ) ) )
12 rnss 5090 . . . . 5  |-  ( ( 1st  |`  ran  F ) 
C_  ( 1st  |`  ( B  X.  C ) )  ->  ran  ( 1st  |` 
ran  F )  C_  ran  ( 1st  |`  ( B  X.  C ) ) )
1310, 11, 123syl 19 . . . 4  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 1st  |`  ran  F
)  C_  ran  ( 1st  |`  ( B  X.  C
) ) )
14 f1stres 6360 . . . . 5  |-  ( 1st  |`  ( B  X.  C
) ) : ( B  X.  C ) --> B
15 frn 5589 . . . . 5  |-  ( ( 1st  |`  ( B  X.  C ) ) : ( B  X.  C
) --> B  ->  ran  ( 1st  |`  ( B  X.  C ) )  C_  B )
1614, 15ax-mp 8 . . . 4  |-  ran  ( 1st  |`  ( B  X.  C ) )  C_  B
1713, 16syl6ss 3352 . . 3  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 1st  |`  ran  F
)  C_  B )
189, 17syl5eqss 3384 . 2  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 1st  o.  F
)  C_  B )
19 df-f 5450 . 2  |-  ( ( 1st  o.  F ) : A --> B  <->  ( ( 1st  o.  F )  Fn  A  /\  ran  ( 1st  o.  F )  C_  B ) )
208, 18, 19sylanbrc 646 1  |-  ( F : A --> ( B  X.  C )  -> 
( 1st  o.  F
) : A --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   _Vcvv 2948    C_ wss 3312    X. cxp 4868   ran crn 4871    |` cres 4872    o. ccom 4874    Fn wfn 5441   -->wf 5442   -onto->wfo 5444   1stc1st 6339
This theorem is referenced by:  ruclem11  12831  ruclem12  12832  caubl  19252
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-13 1727  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  ax-un 4693
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-sbc 3154  df-csb 3244  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-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-1st 6341
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