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Theorem 1stcof 6315
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 6307 . . . 4  |-  1st : _V -onto-> _V
2 fofn 5597 . . . 4  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 8 . . 3  |-  1st  Fn  _V
4 ffn 5533 . . . 4  |-  ( F : A --> ( B  X.  C )  ->  F  Fn  A )
5 dffn2 5534 . . . 4  |-  ( F  Fn  A  <->  F : A
--> _V )
64, 5sylib 189 . . 3  |-  ( F : A --> ( B  X.  C )  ->  F : A --> _V )
7 fnfco 5551 . . 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 5318 . . 3  |-  ran  ( 1st  o.  F )  =  ran  ( 1st  |`  ran  F
)
10 frn 5539 . . . . 5  |-  ( F : A --> ( B  X.  C )  ->  ran  F  C_  ( B  X.  C ) )
11 ssres2 5115 . . . . 5  |-  ( ran 
F  C_  ( B  X.  C )  ->  ( 1st  |`  ran  F ) 
C_  ( 1st  |`  ( B  X.  C ) ) )
12 rnss 5040 . . . . 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 6309 . . . . 5  |-  ( 1st  |`  ( B  X.  C
) ) : ( B  X.  C ) --> B
15 frn 5539 . . . . 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 3305 . . 3  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 1st  |`  ran  F
)  C_  B )
189, 17syl5eqss 3337 . 2  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 1st  o.  F
)  C_  B )
19 df-f 5400 . 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 2901    C_ wss 3265    X. cxp 4818   ran crn 4821    |` cres 4822    o. ccom 4824    Fn wfn 5391   -->wf 5392   -onto->wfo 5394   1stc1st 6288
This theorem is referenced by:  ruclem11  12768  ruclem12  12769  caubl  19133
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fo 5402  df-fv 5404  df-1st 6290
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