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Theorem 1stcof 5732
Description: Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.)
Assertion
Ref Expression
1stcof  F : -->  X.  C  1st  o.  F : -->

Proof of Theorem 1stcof
StepHypRef Expression
1 fo1st 5726 . . . 4  1st : _V -onto-> _V
2 fofn 5051 . . . 4  1st
: _V -onto-> _V  1st 
Fn  _V
31, 2ax-mp 7 . . 3  1st  Fn  _V
4 ffn 4989 . . . 4  F : -->  X.  C 
F  Fn
5 dffn2 4990 . . . 4  F  Fn  F :
--> _V
64, 5sylib 127 . . 3  F : -->  X.  C 
F : --> _V
7 fnfco 5008 . . 3  1st  Fn  _V  F : --> _V  1st  o.  F  Fn
83, 6, 7sylancr 393 . 2  F : -->  X.  C  1st  o.  F  Fn
9 rnco 4770 . . 3  ran  1st  o.  F 
ran  1st  |`  ran  F
10 frn 4995 . . . . 5  F : -->  X.  C 
ran  F  C_  X.  C
11 ssres2 4581 . . . . 5  ran 
F  C_  X.  C  1st  |`  ran  F 
C_  1st  |`  X.  C
12 rnss 4507 . . . . 5  1st  |`  ran  F 
C_  1st  |`  X.  C  ran  1st  |`  ran  F  C_  ran  1st  |`  X.  C
1310, 11, 123syl 17 . . . 4  F : -->  X.  C 
ran  1st  |`  ran  F  C_  ran  1st  |`  X.  C
14 f1stres 5728 . . . . 5  1st  |`  X.  C :  X.  C -->
15 frn 4995 . . . . 5  1st  |`  X.  C :  X.  C
-->  ran  1st  |`  X.  C  C_
1614, 15ax-mp 7 . . . 4  ran  1st  |`  X.  C  C_
1713, 16syl6ss 2951 . . 3  F : -->  X.  C 
ran  1st  |`  ran  F  C_
189, 17syl5eqss 2983 . 2  F : -->  X.  C 
ran  1st  o.  F  C_
19 df-f 4849 . 2  1st  o.  F : -->  1st  o.  F  Fn  ran  1st  o.  F 
C_
208, 18, 19sylanbrc 394 1  F : -->  X.  C  1st  o.  F : -->
Colors of variables: wff set class
Syntax hints:   wi 4   _Vcvv 2551    C_ wss 2911    X. cxp 4286   ran crn 4289    |` cres 4290    o. ccom 4292    Fn wfn 4840   -->wf 4841   -onto->wfo 4843   1stc1st 5707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709
This theorem is referenced by: (None)
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