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Theorem 2ndcof 6718
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 6710 . . . 4  |-  2nd : _V -onto-> _V
2 fofn 5733 . . . 4  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
31, 2ax-mp 5 . . 3  |-  2nd  Fn  _V
4 ffn 5670 . . . 4  |-  ( F : A --> ( B  X.  C )  ->  F  Fn  A )
5 dffn2 5671 . . . 4  |-  ( F  Fn  A  <->  F : A
--> _V )
64, 5sylib 196 . . 3  |-  ( F : A --> ( B  X.  C )  ->  F : A --> _V )
7 fnfco 5688 . . 3  |-  ( ( 2nd  Fn  _V  /\  F : A --> _V )  ->  ( 2nd  o.  F
)  Fn  A )
83, 6, 7sylancr 663 . 2  |-  ( F : A --> ( B  X.  C )  -> 
( 2nd  o.  F
)  Fn  A )
9 rnco 5455 . . 3  |-  ran  ( 2nd  o.  F )  =  ran  ( 2nd  |`  ran  F
)
10 frn 5676 . . . . 5  |-  ( F : A --> ( B  X.  C )  ->  ran  F  C_  ( B  X.  C ) )
11 ssres2 5248 . . . . 5  |-  ( ran 
F  C_  ( B  X.  C )  ->  ( 2nd  |`  ran  F ) 
C_  ( 2nd  |`  ( B  X.  C ) ) )
12 rnss 5179 . . . . 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 6712 . . . . 5  |-  ( 2nd  |`  ( B  X.  C
) ) : ( B  X.  C ) --> C
15 frn 5676 . . . . 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 3479 . . 3  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 2nd  |`  ran  F
)  C_  C )
189, 17syl5eqss 3511 . 2  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 2nd  o.  F
)  C_  C )
19 df-f 5533 . 2  |-  ( ( 2nd  o.  F ) : A --> C  <->  ( ( 2nd  o.  F )  Fn  A  /\  ran  ( 2nd  o.  F )  C_  C ) )
208, 18, 19sylanbrc 664 1  |-  ( F : A --> ( B  X.  C )  -> 
( 2nd  o.  F
) : A --> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   _Vcvv 3078    C_ wss 3439    X. cxp 4949   ran crn 4952    |` cres 4953    o. ccom 4955    Fn wfn 5524   -->wf 5525   -onto->wfo 5527   2ndc2nd 6689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fo 5535  df-fv 5537  df-2nd 6691
This theorem is referenced by:  axdc4lem  8739
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