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Theorem cofunexg 5900
Description: Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
Assertion
Ref Expression
cofunexg  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  o.  B )  e.  _V )

Proof of Theorem cofunexg
StepHypRef Expression
1 relco 5310 . . 3  |-  Rel  ( A  o.  B )
2 relssdmrn 5332 . . 3  |-  ( Rel  ( A  o.  B
)  ->  ( A  o.  B )  C_  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
) )
31, 2ax-mp 8 . 2  |-  ( A  o.  B )  C_  ( dom  ( A  o.  B )  X.  ran  ( A  o.  B
) )
4 dmcoss 5077 . . . . 5  |-  dom  ( A  o.  B )  C_ 
dom  B
5 dmexg 5072 . . . . 5  |-  ( B  e.  C  ->  dom  B  e.  _V )
6 ssexg 4292 . . . . 5  |-  ( ( dom  ( A  o.  B )  C_  dom  B  /\  dom  B  e. 
_V )  ->  dom  ( A  o.  B
)  e.  _V )
74, 5, 6sylancr 645 . . . 4  |-  ( B  e.  C  ->  dom  ( A  o.  B
)  e.  _V )
87adantl 453 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  ( A  o.  B
)  e.  _V )
9 rnco 5318 . . . 4  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
10 rnexg 5073 . . . . . 6  |-  ( B  e.  C  ->  ran  B  e.  _V )
11 resfunexg 5898 . . . . . 6  |-  ( ( Fun  A  /\  ran  B  e.  _V )  -> 
( A  |`  ran  B
)  e.  _V )
1210, 11sylan2 461 . . . . 5  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  ran  B )  e.  _V )
13 rnexg 5073 . . . . 5  |-  ( ( A  |`  ran  B )  e.  _V  ->  ran  ( A  |`  ran  B
)  e.  _V )
1412, 13syl 16 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ran  ( A  |`  ran  B
)  e.  _V )
159, 14syl5eqel 2473 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  ran  ( A  o.  B
)  e.  _V )
16 xpexg 4931 . . 3  |-  ( ( dom  ( A  o.  B )  e.  _V  /\ 
ran  ( A  o.  B )  e.  _V )  ->  ( dom  ( A  o.  B )  X.  ran  ( A  o.  B ) )  e. 
_V )
178, 15, 16syl2anc 643 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
)  e.  _V )
18 ssexg 4292 . 2  |-  ( ( ( A  o.  B
)  C_  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
)  /\  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
)  e.  _V )  ->  ( A  o.  B
)  e.  _V )
193, 17, 18sylancr 645 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  o.  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   _Vcvv 2901    C_ wss 3265    X. cxp 4818   dom cdm 4820   ran crn 4821    |` cres 4822    o. ccom 4824   Rel wrel 4825   Fun wfun 5390
This theorem is referenced by:  cofunex2g  5901  fin1a2lem7  8221  revco  11732  ccatco  11733  isoval  13919  bcthlem4  19151  sinccvglem  24890
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404
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