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Theorem 1stcof 6029
Description: Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
Assertion
Ref Expression
1stcof  |-  ( F : A --> ( B  X.  C )  -> 
( 1st  o.  F
) : A --> B )

Proof of Theorem 1stcof
StepHypRef Expression
1 fo1st 6023 . . . 4  |-  1st : _V -onto-> _V
2 fofn 5317 . . . 4  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 5 . . 3  |-  1st  Fn  _V
4 ffn 5242 . . . 4  |-  ( F : A --> ( B  X.  C )  ->  F  Fn  A )
5 dffn2 5244 . . . 4  |-  ( F  Fn  A  <->  F : A
--> _V )
64, 5sylib 121 . . 3  |-  ( F : A --> ( B  X.  C )  ->  F : A --> _V )
7 fnfco 5267 . . 3  |-  ( ( 1st  Fn  _V  /\  F : A --> _V )  ->  ( 1st  o.  F
)  Fn  A )
83, 6, 7sylancr 410 . 2  |-  ( F : A --> ( B  X.  C )  -> 
( 1st  o.  F
)  Fn  A )
9 rnco 5015 . . 3  |-  ran  ( 1st  o.  F )  =  ran  ( 1st  |`  ran  F
)
10 frn 5251 . . . . 5  |-  ( F : A --> ( B  X.  C )  ->  ran  F  C_  ( B  X.  C ) )
11 ssres2 4816 . . . . 5  |-  ( ran 
F  C_  ( B  X.  C )  ->  ( 1st  |`  ran  F ) 
C_  ( 1st  |`  ( B  X.  C ) ) )
12 rnss 4739 . . . . 5  |-  ( ( 1st  |`  ran  F ) 
C_  ( 1st  |`  ( B  X.  C ) )  ->  ran  ( 1st  |` 
ran  F )  C_  ran  ( 1st  |`  ( B  X.  C ) ) )
1310, 11, 123syl 17 . . . 4  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 1st  |`  ran  F
)  C_  ran  ( 1st  |`  ( B  X.  C
) ) )
14 f1stres 6025 . . . . 5  |-  ( 1st  |`  ( B  X.  C
) ) : ( B  X.  C ) --> B
15 frn 5251 . . . . 5  |-  ( ( 1st  |`  ( B  X.  C ) ) : ( B  X.  C
) --> B  ->  ran  ( 1st  |`  ( B  X.  C ) )  C_  B )
1614, 15ax-mp 5 . . . 4  |-  ran  ( 1st  |`  ( B  X.  C ) )  C_  B
1713, 16sstrdi 3079 . . 3  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 1st  |`  ran  F
)  C_  B )
189, 17eqsstrid 3113 . 2  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 1st  o.  F
)  C_  B )
19 df-f 5097 . 2  |-  ( ( 1st  o.  F ) : A --> B  <->  ( ( 1st  o.  F )  Fn  A  /\  ran  ( 1st  o.  F )  C_  B ) )
208, 18, 19sylanbrc 413 1  |-  ( F : A --> ( B  X.  C )  -> 
( 1st  o.  F
) : A --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   _Vcvv 2660    C_ wss 3041    X. cxp 4507   ran crn 4510    |` cres 4511    o. ccom 4513    Fn wfn 5088   -->wf 5089   -onto->wfo 5091   1stc1st 6004
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101  df-1st 6006
This theorem is referenced by: (None)
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