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Theorem coexg 5168
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 5146 . 2  |-  ( A  o.  B )  C_  ( dom  B  X.  ran  A )
2 dmexg 4886 . . 3  |-  ( B  e.  W  ->  dom  B  e.  _V )
3 rnexg 4887 . . 3  |-  ( A  e.  V  ->  ran  A  e.  _V )
4 xpexg 4736 . . 3  |-  ( ( dom  B  e.  _V  /\ 
ran  A  e.  _V )  ->  ( dom  B  X.  ran  A )  e. 
_V )
52, 3, 4syl2anr 290 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( dom  B  X.  ran  A )  e.  _V )
6 ssexg 4139 . 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 414 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  o.  B
)  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2148   _Vcvv 2737    C_ wss 3129    X. cxp 4620   dom cdm 4622   ran crn 4623    o. ccom 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633
This theorem is referenced by:  coex  5169  climcncf  13704
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