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Theorem coexg 5051
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 5029 . 2  |-  ( A  o.  B )  C_  ( dom  B  X.  ran  A )
2 dmexg 4771 . . 3  |-  ( B  e.  W  ->  dom  B  e.  _V )
3 rnexg 4772 . . 3  |-  ( A  e.  V  ->  ran  A  e.  _V )
4 xpexg 4621 . . 3  |-  ( ( dom  B  e.  _V  /\ 
ran  A  e.  _V )  ->  ( dom  B  X.  ran  A )  e. 
_V )
52, 3, 4syl2anr 286 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( dom  B  X.  ran  A )  e.  _V )
6 ssexg 4035 . 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 408 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  o.  B
)  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1463   _Vcvv 2658    C_ wss 3039    X. cxp 4505   dom cdm 4507   ran crn 4508    o. ccom 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518
This theorem is referenced by:  coex  5052  climcncf  12646
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