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Theorem coexg 5231
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 relco 5187 . . 3  |-  Rel  ( A  o.  B )
2 relssdmrn 5209 . . . 4  |-  ( Rel  ( A  o.  B
)  ->  ( A  o.  B )  C_  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
) )
3 dmcoss 4960 . . . . 5  |-  dom  ( A  o.  B )  C_ 
dom  B
4 rncoss 4961 . . . . 5  |-  ran  ( A  o.  B )  C_ 
ran  A
5 xpss12 4808 . . . . 5  |-  ( ( dom  ( A  o.  B )  C_  dom  B  /\  ran  ( A  o.  B )  C_  ran  A )  ->  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
)  C_  ( dom  B  X.  ran  A ) )
63, 4, 5mp2an 653 . . . 4  |-  ( dom  ( A  o.  B
)  X.  ran  ( A  o.  B )
)  C_  ( dom  B  X.  ran  A )
72, 6syl6ss 3204 . . 3  |-  ( Rel  ( A  o.  B
)  ->  ( A  o.  B )  C_  ( dom  B  X.  ran  A
) )
81, 7ax-mp 8 . 2  |-  ( A  o.  B )  C_  ( dom  B  X.  ran  A )
9 dmexg 4955 . . 3  |-  ( B  e.  W  ->  dom  B  e.  _V )
10 rnexg 4956 . . 3  |-  ( A  e.  V  ->  ran  A  e.  _V )
11 xpexg 4816 . . 3  |-  ( ( dom  B  e.  _V  /\ 
ran  A  e.  _V )  ->  ( dom  B  X.  ran  A )  e. 
_V )
129, 10, 11syl2anr 464 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( dom  B  X.  ran  A )  e.  _V )
13 ssexg 4176 . 2  |-  ( ( ( A  o.  B
)  C_  ( dom  B  X.  ran  A )  /\  ( dom  B  X.  ran  A )  e. 
_V )  ->  ( A  o.  B )  e.  _V )
148, 12, 13sylancr 644 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  o.  B
)  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   _Vcvv 2801    C_ wss 3165    X. cxp 4703   dom cdm 4705   ran crn 4706    o. ccom 4709   Rel wrel 4710
This theorem is referenced by:  coex  5232  wemapwe  7416  cofsmo  7911  supcvg  12330  imasle  13441  setcco  13931  pwsco1mhm  14462  pwsco2mhm  14463  symgov  14793  symgcl  14794  gsumval3  15207  tngds  18180  climcncf  18420  relexpsucr  24041  ov2gc  25226  ov4gc  25227  mapmapmap  25251  injsurinj  25252  f1lindf  27395  mendmulr  27599  climexp  27834  stoweidlem27  27879  stoweidlem31  27883  stoweidlem59  27911  tgrpov  31559  erngmul  31617  erngmul-rN  31625  dvamulr  31823  dvavadd  31826  dvhmulr  31898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716
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