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Theorem preoref22 25229
Description: A preset is reflexive. (Contributed by FL, 22-May-2011.)
Hypothesis
Ref Expression
preoref12.1  |-  X  =  dom  R
Assertion
Ref Expression
preoref22  |-  ( ( R  e. PresetRel  /\  A  e.  X )  ->  A R A )

Proof of Theorem preoref22
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 preodom2 25226 . . 3  |-  ( R  e. PresetRel  ->  dom  R  =  U. U. R )
2 preoref12.1 . . . 4  |-  X  =  dom  R
32preoref12 25228 . . 3  |-  ( R  e. PresetRel  ->  (  _I  |`  X ) 
C_  R )
4 eqtr 2300 . . . . 5  |-  ( ( X  =  dom  R  /\  dom  R  =  U. U. R )  ->  X  =  U. U. R )
5 idref 5759 . . . . . . 7  |-  ( (  _I  |`  U. U. R
)  C_  R  <->  A. x  e.  U. U. R x R x )
6 breq12 4028 . . . . . . . . 9  |-  ( ( x  =  A  /\  x  =  A )  ->  ( x R x  <-> 
A R A ) )
76anidms 626 . . . . . . . 8  |-  ( x  =  A  ->  (
x R x  <->  A R A ) )
87rspccv 2881 . . . . . . 7  |-  ( A. x  e.  U. U. R x R x  ->  ( A  e.  U. U. R  ->  A R A ) )
95, 8sylbi 187 . . . . . 6  |-  ( (  _I  |`  U. U. R
)  C_  R  ->  ( A  e.  U. U. R  ->  A R A ) )
10 reseq2 4950 . . . . . . . 8  |-  ( X  =  U. U. R  ->  (  _I  |`  X )  =  (  _I  |`  U. U. R ) )
1110sseq1d 3205 . . . . . . 7  |-  ( X  =  U. U. R  ->  ( (  _I  |`  X ) 
C_  R  <->  (  _I  |` 
U. U. R )  C_  R ) )
12 eleq2 2344 . . . . . . . 8  |-  ( X  =  U. U. R  ->  ( A  e.  X  <->  A  e.  U. U. R
) )
1312imbi1d 308 . . . . . . 7  |-  ( X  =  U. U. R  ->  ( ( A  e.  X  ->  A R A )  <->  ( A  e.  U. U. R  ->  A R A ) ) )
1411, 13imbi12d 311 . . . . . 6  |-  ( X  =  U. U. R  ->  ( ( (  _I  |`  X )  C_  R  ->  ( A  e.  X  ->  A R A ) )  <->  ( (  _I  |`  U. U. R ) 
C_  R  ->  ( A  e.  U. U. R  ->  A R A ) ) ) )
159, 14mpbiri 224 . . . . 5  |-  ( X  =  U. U. R  ->  ( (  _I  |`  X ) 
C_  R  ->  ( A  e.  X  ->  A R A ) ) )
164, 15syl 15 . . . 4  |-  ( ( X  =  dom  R  /\  dom  R  =  U. U. R )  ->  (
(  _I  |`  X ) 
C_  R  ->  ( A  e.  X  ->  A R A ) ) )
172, 16mpan 651 . . 3  |-  ( dom 
R  =  U. U. R  ->  ( (  _I  |`  X )  C_  R  ->  ( A  e.  X  ->  A R A ) ) )
181, 3, 17sylc 56 . 2  |-  ( R  e. PresetRel  ->  ( A  e.  X  ->  A R A ) )
1918imp 418 1  |-  ( ( R  e. PresetRel  /\  A  e.  X )  ->  A R A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   U.cuni 3827   class class class wbr 4023    _I cid 4304   dom cdm 4689    |` cres 4691  PresetRelcpresetrel 25215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-prs 25223
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