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Theorem sofld 5384
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 5045 . . . . . . . . 9  |-  Rel  ( A  X.  A )
2 relss 5025 . . . . . . . . 9  |-  ( R 
C_  ( A  X.  A )  ->  ( Rel  ( A  X.  A
)  ->  Rel  R ) )
31, 2mpi 17 . . . . . . . 8  |-  ( R 
C_  ( A  X.  A )  ->  Rel  R )
43ad2antlr 726 . . . . . . 7  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  -.  A  C_  ( dom  R  u.  ran  R ) )  ->  Rel  R )
5 df-br 4391 . . . . . . . . . 10  |-  ( x R y  <->  <. x ,  y >.  e.  R
)
6 ssun1 3617 . . . . . . . . . . . . 13  |-  A  C_  ( A  u.  { x } )
7 undif1 3852 . . . . . . . . . . . . 13  |-  ( ( A  \  { x } )  u.  {
x } )  =  ( A  u.  {
x } )
86, 7sseqtr4i 3487 . . . . . . . . . . . 12  |-  A  C_  ( ( A  \  { x } )  u.  { x }
)
9 simpll 753 . . . . . . . . . . . . . 14  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  R  Or  A )
10 dmss 5137 . . . . . . . . . . . . . . . . 17  |-  ( R 
C_  ( A  X.  A )  ->  dom  R 
C_  dom  ( A  X.  A ) )
11 dmxpid 5157 . . . . . . . . . . . . . . . . 17  |-  dom  ( A  X.  A )  =  A
1210, 11syl6sseq 3500 . . . . . . . . . . . . . . . 16  |-  ( R 
C_  ( A  X.  A )  ->  dom  R 
C_  A )
1312ad2antlr 726 . . . . . . . . . . . . . . 15  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  dom  R  C_  A )
143ad2antlr 726 . . . . . . . . . . . . . . . 16  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  Rel  R )
15 releldm 5170 . . . . . . . . . . . . . . . 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 3455 . . . . . . . . . . . . . 14  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  x  e.  A )
18 sossfld 5383 . . . . . . . . . . . . . 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 3617 . . . . . . . . . . . . . . 15  |-  dom  R  C_  ( dom  R  u.  ran  R )
2120, 16sseldi 3452 . . . . . . . . . . . . . 14  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  x  e.  ( dom  R  u.  ran  R ) )
2221snssd 4116 . . . . . . . . . . . . 13  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  { x }  C_  ( dom  R  u.  ran  R ) )
2319, 22unssd 3630 . . . . . . . . . . . 12  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  x R y )  ->  ( ( A  \  { x }
)  u.  { x } )  C_  ( dom  R  u.  ran  R
) )
248, 23syl5ss 3465 . . . . . . . . . . 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 218 . . . . . . . . 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 106 . . . . . . 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 5030 . . . . . 6  |-  ( ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  /\  -.  A  C_  ( dom  R  u.  ran  R ) )  ->  R  C_  (/) )
30 ss0 3766 . . . . . 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 434 . . . 4  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  -> 
( -.  A  C_  ( dom  R  u.  ran  R )  ->  R  =  (/) ) )
3332necon1ad 2664 . . 3  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A ) )  -> 
( R  =/=  (/)  ->  A  C_  ( dom  R  u.  ran  R ) ) )
34333impia 1185 . 2  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A )  /\  R  =/=  (/) )  ->  A  C_  ( dom  R  u.  ran  R ) )
35 rnss 5166 . . . . 5  |-  ( R 
C_  ( A  X.  A )  ->  ran  R 
C_  ran  ( A  X.  A ) )
36 rnxpid 5369 . . . . 5  |-  ran  ( A  X.  A )  =  A
3735, 36syl6sseq 3500 . . . 4  |-  ( R 
C_  ( A  X.  A )  ->  ran  R 
C_  A )
3812, 37unssd 3630 . . 3  |-  ( R 
C_  ( A  X.  A )  ->  ( dom  R  u.  ran  R
)  C_  A )
39383ad2ant2 1010 . 2  |-  ( ( R  Or  A  /\  R  C_  ( A  X.  A )  /\  R  =/=  (/) )  ->  ( dom  R  u.  ran  R
)  C_  A )
4034, 39eqssd 3471 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 965    = wceq 1370    e. wcel 1758    =/= wne 2644    \ cdif 3423    u. cun 3424    C_ wss 3426   (/)c0 3735   {csn 3975   <.cop 3981   class class class wbr 4390    Or wor 4738    X. cxp 4936   dom cdm 4938   ran crn 4939   Rel wrel 4943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-br 4391  df-opab 4449  df-po 4739  df-so 4740  df-xp 4944  df-rel 4945  df-cnv 4946  df-dm 4948  df-rn 4949
This theorem is referenced by: (None)
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