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Theorem mxlelt 25367
Description: The maximal elements of the preset  R. (Contributed by FL, 16-May-2011.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
mxlelt.1  |-  X  = 
U. U. R
Assertion
Ref Expression
mxlelt  |-  ( R  e.  S  ->  ( mxl `  R )  =  { a  e.  X  |  A. b  e.  X  ( a R b  ->  a  =  b ) } )
Distinct variable groups:    R, a,
b    X, a
Allowed substitution hints:    S( a, b)    X( b)

Proof of Theorem mxlelt
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 elex 2809 . 2  |-  ( R  e.  S  ->  R  e.  _V )
2 mxlelt.1 . . . 4  |-  X  = 
U. U. R
32fldrels 25216 . . 3  |-  ( R  e.  S  ->  X  e.  _V )
4 rabexg 4180 . . 3  |-  ( X  e.  _V  ->  { a  e.  X  |  A. b  e.  X  (
a R b  -> 
a  =  b ) }  e.  _V )
53, 4syl 15 . 2  |-  ( R  e.  S  ->  { a  e.  X  |  A. b  e.  X  (
a R b  -> 
a  =  b ) }  e.  _V )
6 unieq 3852 . . . . . 6  |-  ( r  =  R  ->  U. r  =  U. R )
76unieqd 3854 . . . . 5  |-  ( r  =  R  ->  U. U. r  =  U. U. R
)
87, 2syl6eqr 2346 . . . 4  |-  ( r  =  R  ->  U. U. r  =  X )
98eleq2d 2363 . . . . . 6  |-  ( r  =  R  ->  (
b  e.  U. U. r 
<->  b  e.  X ) )
10 breq 4041 . . . . . . 7  |-  ( r  =  R  ->  (
a r b  <->  a R
b ) )
1110imbi1d 308 . . . . . 6  |-  ( r  =  R  ->  (
( a r b  ->  a  =  b )  <->  ( a R b  ->  a  =  b ) ) )
129, 11imbi12d 311 . . . . 5  |-  ( r  =  R  ->  (
( b  e.  U. U. r  ->  ( a
r b  ->  a  =  b ) )  <-> 
( b  e.  X  ->  ( a R b  ->  a  =  b ) ) ) )
1312ralbidv2 2578 . . . 4  |-  ( r  =  R  ->  ( A. b  e.  U. U. r ( a r b  ->  a  =  b )  <->  A. b  e.  X  ( a R b  ->  a  =  b ) ) )
148, 13rabeqbidv 2796 . . 3  |-  ( r  =  R  ->  { a  e.  U. U. r  |  A. b  e.  U. U. r ( a r b  ->  a  =  b ) }  =  { a  e.  X  |  A. b  e.  X  ( a R b  ->  a  =  b ) } )
15 df-mxl 25349 . . 3  |-  mxl  =  ( r  e.  _V  |->  { a  e.  U. U. r  |  A. b  e.  U. U. r ( a r b  -> 
a  =  b ) } )
1614, 15fvmptg 5616 . 2  |-  ( ( R  e.  _V  /\  { a  e.  X  |  A. b  e.  X  ( a R b  ->  a  =  b ) }  e.  _V )  ->  ( mxl `  R
)  =  { a  e.  X  |  A. b  e.  X  (
a R b  -> 
a  =  b ) } )
171, 5, 16syl2anc 642 1  |-  ( R  e.  S  ->  ( mxl `  R )  =  { a  e.  X  |  A. b  e.  X  ( a R b  ->  a  =  b ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801   U.cuni 3843   class class class wbr 4039   ` cfv 5271   mxlcmxl 25319
This theorem is referenced by:  mxlelt2  25368
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-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-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-mxl 25349
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