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Theorem sofld 5319
Description: The base set of a nonempty strict order is the same as the field of the relation. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
sofld  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A )  /\  R  =/=  (/) )  ->  A  =  ( dom  R  u.  ran  R ) )

Proof of Theorem sofld
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 4984 . . . . . . . . 9  |-  Rel  ( A  X.  A )
2 relss 4964 . . . . . . . . 9  |-  ( R 
C_  ( A  X.  A )  ->  ( Rel  ( A  X.  A
)  ->  Rel  R ) )
31, 2mpi 17 . . . . . . . 8  |-  ( R 
C_  ( A  X.  A )  ->  Rel  R )
43ad2antlr 709 . . . . . . 7  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  -.  A  C_  ( dom  R  u.  ran  R ) )  ->  Rel  R )
5 df-br 4214 . . . . . . . . . 10  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
6 ssun1 3511 . . . . . . . . . . . . 13  |-  A  C_  ( A  u.  { x } )
7 undif1 3704 . . . . . . . . . . . . 13  |-  ( ( A  \  { x } )  u.  {
x } )  =  ( A  u.  {
x } )
86, 7sseqtr4i 3382 . . . . . . . . . . . 12  |-  A  C_  ( ( A  \  { x } )  u.  { x }
)
9 simpll 732 . . . . . . . . . . . . . 14  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  R  Or  A )
10 dmss 5070 . . . . . . . . . . . . . . . . 17  |-  ( R 
C_  ( A  X.  A )  ->  dom  R 
C_  dom  ( A  X.  A ) )
11 dmxpid 5090 . . . . . . . . . . . . . . . . 17  |-  dom  ( A  X.  A )  =  A
1210, 11syl6sseq 3395 . . . . . . . . . . . . . . . 16  |-  ( R 
C_  ( A  X.  A )  ->  dom  R 
C_  A )
1312ad2antlr 709 . . . . . . . . . . . . . . 15  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  dom  R  C_  A )
143ad2antlr 709 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  Rel  R )
15 releldm 5103 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  R  /\  x R y )  ->  x  e.  dom  R )
1614, 15sylancom 650 . . . . . . . . . . . . . . 15  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  x  e.  dom  R )
1713, 16sseldd 3350 . . . . . . . . . . . . . 14  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  x  e.  A )
18 sossfld 5318 . . . . . . . . . . . . . 14  |-  ( ( R  Or  A  /\  x  e.  A )  ->  ( A  \  {
x } )  C_  ( dom  R  u.  ran  R ) )
199, 17, 18syl2anc 644 . . . . . . . . . . . . 13  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  ( A  \  { x } ) 
C_  ( dom  R  u.  ran  R ) )
20 ssun1 3511 . . . . . . . . . . . . . . 15  |-  dom  R  C_  ( dom  R  u.  ran  R )
2120, 16sseldi 3347 . . . . . . . . . . . . . 14  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  x  e.  ( dom  R  u.  ran  R ) )
2221snssd 3944 . . . . . . . . . . . . 13  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  { x }  C_  ( dom  R  u.  ran  R ) )
2319, 22unssd 3524 . . . . . . . . . . . 12  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  ( ( A  \  { x }
)  u.  { x } )  C_  ( dom  R  u.  ran  R
) )
248, 23syl5ss 3360 . . . . . . . . . . 11  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  A  C_  ( dom  R  u.  ran  R
) )
2524ex 425 . . . . . . . . . 10  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  -> 
( x R y  ->  A  C_  ( dom  R  u.  ran  R
) ) )
265, 25syl5bir 211 . . . . . . . . 9  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  -> 
( <. x ,  y
>.  e.  R  ->  A  C_  ( dom  R  u.  ran  R ) ) )
2726con3and 430 . . . . . . . 8  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  -.  A  C_  ( dom  R  u.  ran  R ) )  ->  -.  <.
x ,  y >.  e.  R )
2827pm2.21d 101 . . . . . . 7  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  -.  A  C_  ( dom  R  u.  ran  R ) )  ->  ( <. x ,  y >.  e.  R  ->  <. x ,  y >.  e.  (/) ) )
294, 28relssdv 4969 . . . . . 6  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  -.  A  C_  ( dom  R  u.  ran  R ) )  ->  R  C_  (/) )
30 ss0 3659 . . . . . 6  |-  ( R 
C_  (/)  ->  R  =  (/) )
3129, 30syl 16 . . . . 5  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  -.  A  C_  ( dom  R  u.  ran  R ) )  ->  R  =  (/) )
3231ex 425 . . . 4  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  -> 
( -.  A  C_  ( dom  R  u.  ran  R )  ->  R  =  (/) ) )
3332necon1ad 2672 . . 3  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  -> 
( R  =/=  (/)  ->  A  C_  ( dom  R  u.  ran  R ) ) )
34333impia 1151 . 2  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A )  /\  R  =/=  (/) )  ->  A  C_  ( dom  R  u.  ran  R ) )
35 rnss 5099 . . . . 5  |-  ( R 
C_  ( A  X.  A )  ->  ran  R 
C_  ran  ( A  X.  A ) )
36 rnxpid 5303 . . . . 5  |-  ran  ( A  X.  A )  =  A
3735, 36syl6sseq 3395 . . . 4  |-  ( R 
C_  ( A  X.  A )  ->  ran  R 
C_  A )
3812, 37unssd 3524 . . 3  |-  ( R 
C_  ( A  X.  A )  ->  ( dom  R  u.  ran  R
)  C_  A )
39383ad2ant2 980 . 2  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A )  /\  R  =/=  (/) )  ->  ( dom  R  u.  ran  R
)  C_  A )
4034, 39eqssd 3366 1  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A )  /\  R  =/=  (/) )  ->  A  =  ( dom  R  u.  ran  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600    \ cdif 3318    u. cun 3319    C_ wss 3321   (/)c0 3629   {csn 3815   <.cop 3818   class class class wbr 4213    Or wor 4503    X. cxp 4877   dom cdm 4879   ran crn 4880   Rel wrel 4884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-po 4504  df-so 4505  df-xp 4885  df-rel 4886  df-cnv 4887  df-dm 4889  df-rn 4890
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