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Theorem 1stcof 5818
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 5812 . . . 4  |-  1st : _V -onto-> _V
2 fofn 5136 . . . 4  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
31, 2ax-mp 7 . . 3  |-  1st  Fn  _V
4 ffn 5074 . . . 4  |-  ( F : A --> ( B  X.  C )  ->  F  Fn  A )
5 dffn2 5075 . . . 4  |-  ( F  Fn  A  <->  F : A
--> _V )
64, 5sylib 131 . . 3  |-  ( F : A --> ( B  X.  C )  ->  F : A --> _V )
7 fnfco 5093 . . 3  |-  ( ( 1st  Fn  _V  /\  F : A --> _V )  ->  ( 1st  o.  F
)  Fn  A )
83, 6, 7sylancr 399 . 2  |-  ( F : A --> ( B  X.  C )  -> 
( 1st  o.  F
)  Fn  A )
9 rnco 4855 . . 3  |-  ran  ( 1st  o.  F )  =  ran  ( 1st  |`  ran  F
)
10 frn 5080 . . . . 5  |-  ( F : A --> ( B  X.  C )  ->  ran  F  C_  ( B  X.  C ) )
11 ssres2 4666 . . . . 5  |-  ( ran 
F  C_  ( B  X.  C )  ->  ( 1st  |`  ran  F ) 
C_  ( 1st  |`  ( B  X.  C ) ) )
12 rnss 4592 . . . . 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 5814 . . . . 5  |-  ( 1st  |`  ( B  X.  C
) ) : ( B  X.  C ) --> B
15 frn 5080 . . . . 5  |-  ( ( 1st  |`  ( B  X.  C ) ) : ( B  X.  C
) --> B  ->  ran  ( 1st  |`  ( B  X.  C ) )  C_  B )
1614, 15ax-mp 7 . . . 4  |-  ran  ( 1st  |`  ( B  X.  C ) )  C_  B
1713, 16syl6ss 2985 . . 3  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 1st  |`  ran  F
)  C_  B )
189, 17syl5eqss 3017 . 2  |-  ( F : A --> ( B  X.  C )  ->  ran  ( 1st  o.  F
)  C_  B )
19 df-f 4934 . 2  |-  ( ( 1st  o.  F ) : A --> B  <->  ( ( 1st  o.  F )  Fn  A  /\  ran  ( 1st  o.  F )  C_  B ) )
208, 18, 19sylanbrc 402 1  |-  ( F : A --> ( B  X.  C )  -> 
( 1st  o.  F
) : A --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   _Vcvv 2574    C_ wss 2945    X. cxp 4371   ran crn 4374    |` cres 4375    o. ccom 4377    Fn wfn 4925   -->wf 4926   -onto->wfo 4928   1stc1st 5793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938  df-1st 5795
This theorem is referenced by: (None)
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