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Theorem cofunexg 6009
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 5037 . . 3  |-  Rel  ( A  o.  B )
2 relssdmrn 5059 . . 3  |-  ( Rel  ( A  o.  B
)  ->  ( A  o.  B )  C_  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
) )
31, 2ax-mp 5 . 2  |-  ( A  o.  B )  C_  ( dom  ( A  o.  B )  X.  ran  ( A  o.  B
) )
4 dmcoss 4808 . . . . 5  |-  dom  ( A  o.  B )  C_ 
dom  B
5 dmexg 4803 . . . . 5  |-  ( B  e.  C  ->  dom  B  e.  _V )
6 ssexg 4067 . . . . 5  |-  ( ( dom  ( A  o.  B )  C_  dom  B  /\  dom  B  e. 
_V )  ->  dom  ( A  o.  B
)  e.  _V )
74, 5, 6sylancr 410 . . . 4  |-  ( B  e.  C  ->  dom  ( A  o.  B
)  e.  _V )
87adantl 275 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  ( A  o.  B
)  e.  _V )
9 rnco 5045 . . . 4  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
10 rnexg 4804 . . . . . 6  |-  ( B  e.  C  ->  ran  B  e.  _V )
11 resfunexg 5641 . . . . . 6  |-  ( ( Fun  A  /\  ran  B  e.  _V )  -> 
( A  |`  ran  B
)  e.  _V )
1210, 11sylan2 284 . . . . 5  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  ran  B )  e.  _V )
13 rnexg 4804 . . . . 5  |-  ( ( A  |`  ran  B )  e.  _V  ->  ran  ( A  |`  ran  B
)  e.  _V )
1412, 13syl 14 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ran  ( A  |`  ran  B
)  e.  _V )
159, 14eqeltrid 2226 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  ran  ( A  o.  B
)  e.  _V )
16 xpexg 4653 . . 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 408 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
)  e.  _V )
18 ssexg 4067 . 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 410 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  o.  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1480   _Vcvv 2686    C_ wss 3071    X. cxp 4537   dom cdm 4539   ran crn 4540    |` cres 4541    o. ccom 4543   Rel wrel 4544   Fun wfun 5117
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131
This theorem is referenced by:  cofunex2g  6010  ctm  6994  ctssdclemr  6997
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