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Theorem 2ndcof 6377
Description: Composition of the first member function with another function. (Contributed by FL, 15-Oct-2012.)
Assertion
Ref Expression
2ndcof  |-  ( F : A --> ( B  X.  C )  -> 
( 2nd  o.  F
) : A --> C )

Proof of Theorem 2ndcof
StepHypRef Expression
1 fo2nd 6369 . . . 4  |-  2nd : _V -onto-> _V
2 fofn 5657 . . . 4  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
31, 2ax-mp 8 . . 3  |-  2nd  Fn  _V
4 ffn 5593 . . . 4  |-  ( F : A --> ( B  X.  C )  ->  F  Fn  A )
5 dffn2 5594 . . . 4  |-  ( F  Fn  A  <->  F : A
--> _V )
64, 5sylib 190 . . 3  |-  ( F : A --> ( B  X.  C )  ->  F : A --> _V )
7 fnfco 5611 . . 3  |-  ( ( 2nd  Fn  _V  /\  F : A --> _V )  ->  ( 2nd  o.  F
)  Fn  A )
83, 6, 7sylancr 646 . 2  |-  ( F : A --> ( B  X.  C )  -> 
( 2nd  o.  F
)  Fn  A )
9 rnco 5378 . . 3  |-  ran  ( 2nd  o.  F )  =  ran  ( 2nd  |`  ran  F
)
10 frn 5599 . . . . 5  |-  ( F : A --> ( B  X.  C )  ->  ran  F  C_  ( B  X.  C ) )
11 ssres2 5175 . . . . 5  |-  ( ran 
F  C_  ( B  X.  C )  ->  ( 2nd  |`  ran  F ) 
C_  ( 2nd  |`  ( B  X.  C ) ) )
12 rnss 5100 . . . . 5  |-  ( ( 2nd  |`  ran  F ) 
C_  ( 2nd  |`  ( B  X.  C ) )  ->  ran  ( 2nd  |` 
ran  F )  C_  ran  ( 2nd  |`  ( B  X.  C ) ) )
1310, 11, 123syl 19 . . . 4  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 2nd  |`  ran  F
)  C_  ran  ( 2nd  |`  ( B  X.  C
) ) )
14 f2ndres 6371 . . . . 5  |-  ( 2nd  |`  ( B  X.  C
) ) : ( B  X.  C ) --> C
15 frn 5599 . . . . 5  |-  ( ( 2nd  |`  ( B  X.  C ) ) : ( B  X.  C
) --> C  ->  ran  ( 2nd  |`  ( B  X.  C ) )  C_  C )
1614, 15ax-mp 8 . . . 4  |-  ran  ( 2nd  |`  ( B  X.  C ) )  C_  C
1713, 16syl6ss 3362 . . 3  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 2nd  |`  ran  F
)  C_  C )
189, 17syl5eqss 3394 . 2  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 2nd  o.  F
)  C_  C )
19 df-f 5460 . 2  |-  ( ( 2nd  o.  F ) : A --> C  <->  ( ( 2nd  o.  F )  Fn  A  /\  ran  ( 2nd  o.  F )  C_  C ) )
208, 18, 19sylanbrc 647 1  |-  ( F : A --> ( B  X.  C )  -> 
( 2nd  o.  F
) : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4   _Vcvv 2958    C_ wss 3322    X. cxp 4878   ran crn 4881    |` cres 4882    o. ccom 4884    Fn wfn 5451   -->wf 5452   -onto->wfo 5454   2ndc2nd 6350
This theorem is referenced by:  axdc4lem  8337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-2nd 6352
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