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Theorem sofld 5274
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 4933 . . . . . . . . 9  |-  Rel  ( A  X.  A )
2 relss 4913 . . . . . . . . 9  |-  ( R 
C_  ( A  X.  A )  ->  ( Rel  ( A  X.  A
)  ->  Rel  R ) )
31, 2mpi 21 . . . . . . . 8  |-  ( R 
C_  ( A  X.  A )  ->  Rel  R )
43ad2antlr 727 . . . . . . 7  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  -.  A  C_  ( dom  R  u.  ran  R ) )  ->  Rel  R )
5 df-br 4398 . . . . . . . . . 10  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
6 ssun1 3608 . . . . . . . . . . . . 13  |-  A  C_  ( A  u.  { x } )
7 undif1 3849 . . . . . . . . . . . . 13  |-  ( ( A  \  { x } )  u.  {
x } )  =  ( A  u.  {
x } )
86, 7sseqtr4i 3477 . . . . . . . . . . . 12  |-  A  C_  ( ( A  \  { x } )  u.  { x }
)
9 simpll 754 . . . . . . . . . . . . . 14  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  R  Or  A )
10 dmss 5025 . . . . . . . . . . . . . . . . 17  |-  ( R 
C_  ( A  X.  A )  ->  dom  R 
C_  dom  ( A  X.  A ) )
11 dmxpid 5045 . . . . . . . . . . . . . . . . 17  |-  dom  ( A  X.  A )  =  A
1210, 11syl6sseq 3490 . . . . . . . . . . . . . . . 16  |-  ( R 
C_  ( A  X.  A )  ->  dom  R 
C_  A )
1312ad2antlr 727 . . . . . . . . . . . . . . 15  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  dom  R  C_  A )
143ad2antlr 727 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  Rel  R )
15 releldm 5058 . . . . . . . . . . . . . . . 16  |-  ( ( Rel  R  /\  x R y )  ->  x  e.  dom  R )
1614, 15sylancom 667 . . . . . . . . . . . . . . 15  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  x  e.  dom  R )
1713, 16sseldd 3445 . . . . . . . . . . . . . 14  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  x  e.  A )
18 sossfld 5273 . . . . . . . . . . . . . 14  |-  ( ( R  Or  A  /\  x  e.  A )  ->  ( A  \  {
x } )  C_  ( dom  R  u.  ran  R ) )
199, 17, 18syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  ( A  \  { x } ) 
C_  ( dom  R  u.  ran  R ) )
20 ssun1 3608 . . . . . . . . . . . . . . 15  |-  dom  R  C_  ( dom  R  u.  ran  R )
2120, 16sseldi 3442 . . . . . . . . . . . . . 14  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  x  e.  ( dom  R  u.  ran  R ) )
2221snssd 4119 . . . . . . . . . . . . 13  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  { x }  C_  ( dom  R  u.  ran  R ) )
2319, 22unssd 3621 . . . . . . . . . . . 12  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  ( ( A  \  { x }
)  u.  { x } )  C_  ( dom  R  u.  ran  R
) )
248, 23syl5ss 3455 . . . . . . . . . . 11  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  A  C_  ( dom  R  u.  ran  R
) )
2524ex 434 . . . . . . . . . 10  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  -> 
( x R y  ->  A  C_  ( dom  R  u.  ran  R
) ) )
265, 25syl5bir 220 . . . . . . . . 9  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  -> 
( <. x ,  y
>.  e.  R  ->  A  C_  ( dom  R  u.  ran  R ) ) )
2726con3dimp 441 . . . . . . . 8  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  -.  A  C_  ( dom  R  u.  ran  R ) )  ->  -.  <.
x ,  y >.  e.  R )
2827pm2.21d 108 . . . . . . 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 4918 . . . . . 6  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  -.  A  C_  ( dom  R  u.  ran  R ) )  ->  R  C_  (/) )
30 ss0 3772 . . . . . 6  |-  ( R 
C_  (/)  ->  R  =  (/) )
3129, 30syl 17 . . . . 5  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  -.  A  C_  ( dom  R  u.  ran  R ) )  ->  R  =  (/) )
3231ex 434 . . . 4  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  -> 
( -.  A  C_  ( dom  R  u.  ran  R )  ->  R  =  (/) ) )
3332necon1ad 2621 . . 3  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  -> 
( R  =/=  (/)  ->  A  C_  ( dom  R  u.  ran  R ) ) )
34333impia 1196 . 2  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A )  /\  R  =/=  (/) )  ->  A  C_  ( dom  R  u.  ran  R ) )
35 rnss 5054 . . . . 5  |-  ( R 
C_  ( A  X.  A )  ->  ran  R 
C_  ran  ( A  X.  A ) )
36 rnxpid 5260 . . . . 5  |-  ran  ( A  X.  A )  =  A
3735, 36syl6sseq 3490 . . . 4  |-  ( R 
C_  ( A  X.  A )  ->  ran  R 
C_  A )
3812, 37unssd 3621 . . 3  |-  ( R 
C_  ( A  X.  A )  ->  ( dom  R  u.  ran  R
)  C_  A )
39383ad2ant2 1021 . 2  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A )  /\  R  =/=  (/) )  ->  ( dom  R  u.  ran  R
)  C_  A )
4034, 39eqssd 3461 1  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A )  /\  R  =/=  (/) )  ->  A  =  ( dom  R  u.  ran  R ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 976    = wceq 1407    e. wcel 1844    =/= wne 2600    \ cdif 3413    u. cun 3414    C_ wss 3416   (/)c0 3740   {csn 3974   <.cop 3980   class class class wbr 4397    Or wor 4745    X. cxp 4823   dom cdm 4825   ran crn 4826   Rel wrel 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-br 4398  df-opab 4456  df-po 4746  df-so 4747  df-xp 4831  df-rel 4832  df-cnv 4833  df-dm 4835  df-rn 4836
This theorem is referenced by: (None)
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