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Theorem cofunexg 6652
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 5445 . . 3  |-  Rel  ( A  o.  B )
2 relssdmrn 5467 . . 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 5208 . . . . 5  |-  dom  ( A  o.  B )  C_ 
dom  B
5 dmexg 6620 . . . . 5  |-  ( B  e.  C  ->  dom  B  e.  _V )
6 ssexg 4547 . . . . 5  |-  ( ( dom  ( A  o.  B )  C_  dom  B  /\  dom  B  e. 
_V )  ->  dom  ( A  o.  B
)  e.  _V )
74, 5, 6sylancr 663 . . . 4  |-  ( B  e.  C  ->  dom  ( A  o.  B
)  e.  _V )
87adantl 466 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  dom  ( A  o.  B
)  e.  _V )
9 rnco 5453 . . . 4  |-  ran  ( A  o.  B )  =  ran  ( A  |`  ran  B )
10 rnexg 6621 . . . . . 6  |-  ( B  e.  C  ->  ran  B  e.  _V )
11 resfunexg 6053 . . . . . 6  |-  ( ( Fun  A  /\  ran  B  e.  _V )  -> 
( A  |`  ran  B
)  e.  _V )
1210, 11sylan2 474 . . . . 5  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  |`  ran  B )  e.  _V )
13 rnexg 6621 . . . . 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 2546 . . 3  |-  ( ( Fun  A  /\  B  e.  C )  ->  ran  ( A  o.  B
)  e.  _V )
16 xpexg 6618 . . 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 661 . 2  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
)  e.  _V )
18 ssexg 4547 . 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 663 1  |-  ( ( Fun  A  /\  B  e.  C )  ->  ( A  o.  B )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1758   _Vcvv 3078    C_ wss 3437    X. cxp 4947   dom cdm 4949   ran crn 4950    |` cres 4951    o. ccom 4953   Rel wrel 4954   Fun wfun 5521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535
This theorem is referenced by:  cofunex2g  6653  fin1a2lem7  8687  revco  12581  ccatco  12582  lswco  12585  isoval  14823  bcthlem4  20971  sseqval  26916  sinccvglem  27462
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