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Theorem 2ndcof 6594
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 6586 . . . 4  |-  2nd : _V -onto-> _V
2 fofn 5610 . . . 4  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
31, 2ax-mp 5 . . 3  |-  2nd  Fn  _V
4 ffn 5547 . . . 4  |-  ( F : A --> ( B  X.  C )  ->  F  Fn  A )
5 dffn2 5548 . . . 4  |-  ( F  Fn  A  <->  F : A
--> _V )
64, 5sylib 196 . . 3  |-  ( F : A --> ( B  X.  C )  ->  F : A --> _V )
7 fnfco 5565 . . 3  |-  ( ( 2nd  Fn  _V  /\  F : A --> _V )  ->  ( 2nd  o.  F
)  Fn  A )
83, 6, 7sylancr 656 . 2  |-  ( F : A --> ( B  X.  C )  -> 
( 2nd  o.  F
)  Fn  A )
9 rnco 5332 . . 3  |-  ran  ( 2nd  o.  F )  =  ran  ( 2nd  |`  ran  F
)
10 frn 5553 . . . . 5  |-  ( F : A --> ( B  X.  C )  ->  ran  F  C_  ( B  X.  C ) )
11 ssres2 5125 . . . . 5  |-  ( ran 
F  C_  ( B  X.  C )  ->  ( 2nd  |`  ran  F ) 
C_  ( 2nd  |`  ( B  X.  C ) ) )
12 rnss 5055 . . . . 5  |-  ( ( 2nd  |`  ran  F ) 
C_  ( 2nd  |`  ( B  X.  C ) )  ->  ran  ( 2nd  |` 
ran  F )  C_  ran  ( 2nd  |`  ( B  X.  C ) ) )
1310, 11, 123syl 20 . . . 4  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 2nd  |`  ran  F
)  C_  ran  ( 2nd  |`  ( B  X.  C
) ) )
14 f2ndres 6588 . . . . 5  |-  ( 2nd  |`  ( B  X.  C
) ) : ( B  X.  C ) --> C
15 frn 5553 . . . . 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, 16syl6ss 3356 . . 3  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 2nd  |`  ran  F
)  C_  C )
189, 17syl5eqss 3388 . 2  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 2nd  o.  F
)  C_  C )
19 df-f 5410 . 2  |-  ( ( 2nd  o.  F ) : A --> C  <->  ( ( 2nd  o.  F )  Fn  A  /\  ran  ( 2nd  o.  F )  C_  C ) )
208, 18, 19sylanbrc 657 1  |-  ( F : A --> ( B  X.  C )  -> 
( 2nd  o.  F
) : A --> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   _Vcvv 2962    C_ wss 3316    X. cxp 4825   ran crn 4828    |` cres 4829    o. ccom 4831    Fn wfn 5401   -->wf 5402   -onto->wfo 5404   2ndc2nd 6565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-fo 5412  df-fv 5414  df-2nd 6567
This theorem is referenced by:  axdc4lem  8612
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