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Theorem cofunexg 5739
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 5171 . . 3  |-  Rel  ( A  o.  B )
2 relssdmrn 5193 . . 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 4944 . . . . 5  |-  dom  ( A  o.  B )  C_ 
dom  B
5 dmexg 4939 . . . . 5  |-  ( B  e.  C  ->  dom  B  e.  _V )
6 ssexg 4160 . . . . 5  |-  ( ( dom  ( A  o.  B )  C_  dom  B  /\  dom  B  e. 
_V )  ->  dom  ( A  o.  B
)  e.  _V )
74, 5, 6sylancr 644 . . . 4  |-  ( B  e.  C  ->  dom  ( A  o.  B
)  e.  _V )
87adantl 452 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  ( A  o.  B
)  e.  _V )
9 rnco 5179 . . . 4  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
10 rnexg 4940 . . . . . 6  |-  ( B  e.  C  ->  ran  B  e.  _V )
11 resfunexg 5737 . . . . . 6  |-  ( ( Fun  A  /\  ran  B  e.  _V )  -> 
( A  |`  ran  B
)  e.  _V )
1210, 11sylan2 460 . . . . 5  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  ran  B )  e.  _V )
13 rnexg 4940 . . . . 5  |-  ( ( A  |`  ran  B )  e.  _V  ->  ran  ( A  |`  ran  B
)  e.  _V )
1412, 13syl 15 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ran  ( A  |`  ran  B
)  e.  _V )
159, 14syl5eqel 2367 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  ran  ( A  o.  B
)  e.  _V )
16 xpexg 4800 . . 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 642 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
)  e.  _V )
18 ssexg 4160 . 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 644 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  o.  B )  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   _Vcvv 2788    C_ wss 3152    X. cxp 4687   dom cdm 4689   ran crn 4690    |` cres 4691    o. ccom 4693   Rel wrel 4694   Fun wfun 5249
This theorem is referenced by:  cofunex2g  5740  fin1a2lem7  8032  revco  11489  ccatco  11490  isoval  13667  bcthlem4  18749  sinccvglem  24005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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