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Theorem 2ndcof 6148
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 6140 . . . 4  |-  2nd : _V -onto-> _V
2 fofn 5453 . . . 4  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
31, 2ax-mp 8 . . 3  |-  2nd  Fn  _V
4 ffn 5389 . . . 4  |-  ( F : A --> ( B  X.  C )  ->  F  Fn  A )
5 dffn2 5390 . . . 4  |-  ( F  Fn  A  <->  F : A
--> _V )
64, 5sylib 188 . . 3  |-  ( F : A --> ( B  X.  C )  ->  F : A --> _V )
7 fnfco 5407 . . 3  |-  ( ( 2nd  Fn  _V  /\  F : A --> _V )  ->  ( 2nd  o.  F
)  Fn  A )
83, 6, 7sylancr 644 . 2  |-  ( F : A --> ( B  X.  C )  -> 
( 2nd  o.  F
)  Fn  A )
9 rnco 5179 . . 3  |-  ran  ( 2nd  o.  F )  =  ran  ( 2nd  |`  ran  F
)
10 frn 5395 . . . . 5  |-  ( F : A --> ( B  X.  C )  ->  ran  F  C_  ( B  X.  C ) )
11 ssres2 4982 . . . . 5  |-  ( ran 
F  C_  ( B  X.  C )  ->  ( 2nd  |`  ran  F ) 
C_  ( 2nd  |`  ( B  X.  C ) ) )
12 rnss 4907 . . . . 5  |-  ( ( 2nd  |`  ran  F ) 
C_  ( 2nd  |`  ( B  X.  C ) )  ->  ran  ( 2nd  |` 
ran  F )  C_  ran  ( 2nd  |`  ( B  X.  C ) ) )
1310, 11, 123syl 18 . . . 4  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 2nd  |`  ran  F
)  C_  ran  ( 2nd  |`  ( B  X.  C
) ) )
14 f2ndres 6142 . . . . 5  |-  ( 2nd  |`  ( B  X.  C
) ) : ( B  X.  C ) --> C
15 frn 5395 . . . . 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 3191 . . 3  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 2nd  |`  ran  F
)  C_  C )
189, 17syl5eqss 3222 . 2  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 2nd  o.  F
)  C_  C )
19 df-f 5259 . 2  |-  ( ( 2nd  o.  F ) : A --> C  <->  ( ( 2nd  o.  F )  Fn  A  /\  ran  ( 2nd  o.  F )  C_  C ) )
208, 18, 19sylanbrc 645 1  |-  ( F : A --> ( B  X.  C )  -> 
( 2nd  o.  F
) : A --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4   _Vcvv 2788    C_ wss 3152    X. cxp 4687   ran crn 4690    |` cres 4691    o. ccom 4693    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   2ndc2nd 6121
This theorem is referenced by:  axdc4lem  8081  limptlimpr2lem1  25574  limptlimpr2lem2  25575
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-2nd 6123
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