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Theorem coexg 4975
Description: The composition of two sets is a set. (Contributed by NM, 19-Mar-1998.)
Assertion
Ref Expression
coexg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  o.  B
)  e.  _V )

Proof of Theorem coexg
StepHypRef Expression
1 cossxp 4953 . 2  |-  ( A  o.  B )  C_  ( dom  B  X.  ran  A )
2 dmexg 4697 . . 3  |-  ( B  e.  W  ->  dom  B  e.  _V )
3 rnexg 4698 . . 3  |-  ( A  e.  V  ->  ran  A  e.  _V )
4 xpexg 4552 . . 3  |-  ( ( dom  B  e.  _V  /\ 
ran  A  e.  _V )  ->  ( dom  B  X.  ran  A )  e. 
_V )
52, 3, 4syl2anr 284 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( dom  B  X.  ran  A )  e.  _V )
6 ssexg 3978 . 2  |-  ( ( ( A  o.  B
)  C_  ( dom  B  X.  ran  A )  /\  ( dom  B  X.  ran  A )  e. 
_V )  ->  ( A  o.  B )  e.  _V )
71, 5, 6sylancr 405 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  o.  B
)  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1438   _Vcvv 2619    C_ wss 2999    X. cxp 4436   dom cdm 4438   ran crn 4439    o. ccom 4442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449
This theorem is referenced by:  coex  4976  climcncf  11640
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