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Theorem 2ndcof 6115
Description: Composition of the second 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 6109 . . . 4  |-  2nd : _V -onto-> _V
2 fofn 5397 . . . 4  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
31, 2ax-mp 5 . . 3  |-  2nd  Fn  _V
4 ffn 5322 . . . 4  |-  ( F : A --> ( B  X.  C )  ->  F  Fn  A )
5 dffn2 5324 . . . 4  |-  ( F  Fn  A  <->  F : A
--> _V )
64, 5sylib 121 . . 3  |-  ( F : A --> ( B  X.  C )  ->  F : A --> _V )
7 fnfco 5347 . . 3  |-  ( ( 2nd  Fn  _V  /\  F : A --> _V )  ->  ( 2nd  o.  F
)  Fn  A )
83, 6, 7sylancr 411 . 2  |-  ( F : A --> ( B  X.  C )  -> 
( 2nd  o.  F
)  Fn  A )
9 rnco 5095 . . 3  |-  ran  ( 2nd  o.  F )  =  ran  ( 2nd  |`  ran  F
)
10 frn 5331 . . . . 5  |-  ( F : A --> ( B  X.  C )  ->  ran  F  C_  ( B  X.  C ) )
11 ssres2 4896 . . . . 5  |-  ( ran 
F  C_  ( B  X.  C )  ->  ( 2nd  |`  ran  F ) 
C_  ( 2nd  |`  ( B  X.  C ) ) )
12 rnss 4819 . . . . 5  |-  ( ( 2nd  |`  ran  F ) 
C_  ( 2nd  |`  ( B  X.  C ) )  ->  ran  ( 2nd  |` 
ran  F )  C_  ran  ( 2nd  |`  ( B  X.  C ) ) )
1310, 11, 123syl 17 . . . 4  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 2nd  |`  ran  F
)  C_  ran  ( 2nd  |`  ( B  X.  C
) ) )
14 f2ndres 6111 . . . . 5  |-  ( 2nd  |`  ( B  X.  C
) ) : ( B  X.  C ) --> C
15 frn 5331 . . . . 5  |-  ( ( 2nd  |`  ( B  X.  C ) ) : ( B  X.  C
) --> C  ->  ran  ( 2nd  |`  ( B  X.  C ) )  C_  C )
1614, 15ax-mp 5 . . . 4  |-  ran  ( 2nd  |`  ( B  X.  C ) )  C_  C
1713, 16sstrdi 3140 . . 3  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 2nd  |`  ran  F
)  C_  C )
189, 17eqsstrid 3174 . 2  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 2nd  o.  F
)  C_  C )
19 df-f 5177 . 2  |-  ( ( 2nd  o.  F ) : A --> C  <->  ( ( 2nd  o.  F )  Fn  A  /\  ran  ( 2nd  o.  F )  C_  C ) )
208, 18, 19sylanbrc 414 1  |-  ( F : A --> ( B  X.  C )  -> 
( 2nd  o.  F
) : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4   _Vcvv 2712    C_ wss 3102    X. cxp 4587   ran crn 4590    |` cres 4591    o. ccom 4593    Fn wfn 5168   -->wf 5169   -onto->wfo 5171   2ndc2nd 6090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172  ax-un 4396
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-id 4256  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-fo 5179  df-fv 5181  df-2nd 6092
This theorem is referenced by: (None)
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