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Theorem cofunexg 6757
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 5333 . . 3  |-  Rel  ( A  o.  B )
2 relssdmrn 5356 . . 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 5094 . . . . 5  |-  dom  ( A  o.  B )  C_ 
dom  B
5 dmexg 6724 . . . . 5  |-  ( B  e.  C  ->  dom  B  e.  _V )
6 ssexg 4549 . . . . 5  |-  ( ( dom  ( A  o.  B )  C_  dom  B  /\  dom  B  e. 
_V )  ->  dom  ( A  o.  B
)  e.  _V )
74, 5, 6sylancr 669 . . . 4  |-  ( B  e.  C  ->  dom  ( A  o.  B
)  e.  _V )
87adantl 468 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  ( A  o.  B
)  e.  _V )
9 rnco 5341 . . . 4  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
10 rnexg 6725 . . . . . 6  |-  ( B  e.  C  ->  ran  B  e.  _V )
11 resfunexg 6130 . . . . . 6  |-  ( ( Fun  A  /\  ran  B  e.  _V )  -> 
( A  |`  ran  B
)  e.  _V )
1210, 11sylan2 477 . . . . 5  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  ran  B )  e.  _V )
13 rnexg 6725 . . . . 5  |-  ( ( A  |`  ran  B )  e.  _V  ->  ran  ( A  |`  ran  B
)  e.  _V )
1412, 13syl 17 . . . 4  |-  ( ( Fun  A  /\  B  e.  C )  ->  ran  ( A  |`  ran  B
)  e.  _V )
159, 14syl5eqel 2533 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  ran  ( A  o.  B
)  e.  _V )
16 xpexg 6593 . . 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 667 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
)  e.  _V )
18 ssexg 4549 . 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 669 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  o.  B )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    e. wcel 1887   _Vcvv 3045    C_ wss 3404    X. cxp 4832   dom cdm 4834   ran crn 4835    |` cres 4836    o. ccom 4838   Rel wrel 4839   Fun wfun 5576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590
This theorem is referenced by:  cofunex2g  6758  fin1a2lem7  8836  revco  12931  ccatco  12932  lswco  12935  isofval  15662  bcthlem4  22295  sseqval  29221  sinccvglem  30316  pfxco  38979
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