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Theorem pofun 4330
Description: A function preserves a partial order relation. (Contributed by Jeff Madsen, 18-Jun-2011.)
Hypotheses
Ref Expression
pofun.1  |-  S  =  { <. x ,  y
>.  |  X R Y }
pofun.2  |-  ( x  =  y  ->  X  =  Y )
Assertion
Ref Expression
pofun  |-  ( ( R  Po  B  /\  A. x  e.  A  X  e.  B )  ->  S  Po  A )
Distinct variable groups:    x, R, y    y, X    x, Y    x, A    x, B
Allowed substitution hints:    A( y)    B( y)    S( x, y)    X( x)    Y( y)

Proof of Theorem pofun
Dummy variables  v  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcsb1v 3113 . . . . . . 7  |-  F/_ x [_ v  /  x ]_ X
21nfel1 2429 . . . . . 6  |-  F/ x [_ v  /  x ]_ X  e.  B
3 csbeq1a 3089 . . . . . . 7  |-  ( x  =  v  ->  X  =  [_ v  /  x ]_ X )
43eleq1d 2349 . . . . . 6  |-  ( x  =  v  ->  ( X  e.  B  <->  [_ v  /  x ]_ X  e.  B
) )
52, 4rspc 2878 . . . . 5  |-  ( v  e.  A  ->  ( A. x  e.  A  X  e.  B  ->  [_ v  /  x ]_ X  e.  B )
)
65impcom 419 . . . 4  |-  ( ( A. x  e.  A  X  e.  B  /\  v  e.  A )  ->  [_ v  /  x ]_ X  e.  B
)
7 poirr 4325 . . . . 5  |-  ( ( R  Po  B  /\  [_ v  /  x ]_ X  e.  B )  ->  -.  [_ v  /  x ]_ X R [_ v  /  x ]_ X
)
8 df-br 4024 . . . . . 6  |-  ( v S v  <->  <. v ,  v >.  e.  S
)
9 pofun.1 . . . . . . 7  |-  S  =  { <. x ,  y
>.  |  X R Y }
109eleq2i 2347 . . . . . 6  |-  ( <.
v ,  v >.  e.  S  <->  <. v ,  v
>.  e.  { <. x ,  y >.  |  X R Y } )
11 nfcv 2419 . . . . . . . 8  |-  F/_ x R
12 nfcv 2419 . . . . . . . 8  |-  F/_ x Y
131, 11, 12nfbr 4067 . . . . . . 7  |-  F/ x [_ v  /  x ]_ X R Y
14 nfv 1605 . . . . . . 7  |-  F/ y
[_ v  /  x ]_ X R [_ v  /  x ]_ X
15 vex 2791 . . . . . . 7  |-  v  e. 
_V
163breq1d 4033 . . . . . . 7  |-  ( x  =  v  ->  ( X R Y  <->  [_ v  /  x ]_ X R Y ) )
17 vex 2791 . . . . . . . . . 10  |-  y  e. 
_V
18 pofun.2 . . . . . . . . . 10  |-  ( x  =  y  ->  X  =  Y )
1917, 12, 18csbief 3122 . . . . . . . . 9  |-  [_ y  /  x ]_ X  =  Y
20 csbeq1 3084 . . . . . . . . 9  |-  ( y  =  v  ->  [_ y  /  x ]_ X  = 
[_ v  /  x ]_ X )
2119, 20syl5eqr 2329 . . . . . . . 8  |-  ( y  =  v  ->  Y  =  [_ v  /  x ]_ X )
2221breq2d 4035 . . . . . . 7  |-  ( y  =  v  ->  ( [_ v  /  x ]_ X R Y  <->  [_ v  /  x ]_ X R [_ v  /  x ]_ X
) )
2313, 14, 15, 15, 16, 22opelopabf 4289 . . . . . 6  |-  ( <.
v ,  v >.  e.  { <. x ,  y
>.  |  X R Y }  <->  [_ v  /  x ]_ X R [_ v  /  x ]_ X )
248, 10, 233bitri 262 . . . . 5  |-  ( v S v  <->  [_ v  /  x ]_ X R [_ v  /  x ]_ X
)
257, 24sylnibr 296 . . . 4  |-  ( ( R  Po  B  /\  [_ v  /  x ]_ X  e.  B )  ->  -.  v S v )
266, 25sylan2 460 . . 3  |-  ( ( R  Po  B  /\  ( A. x  e.  A  X  e.  B  /\  v  e.  A )
)  ->  -.  v S v )
2726anassrs 629 . 2  |-  ( ( ( R  Po  B  /\  A. x  e.  A  X  e.  B )  /\  v  e.  A
)  ->  -.  v S v )
285com12 27 . . . . . 6  |-  ( A. x  e.  A  X  e.  B  ->  ( v  e.  A  ->  [_ v  /  x ]_ X  e.  B ) )
29 nfcsb1v 3113 . . . . . . . . 9  |-  F/_ x [_ w  /  x ]_ X
3029nfel1 2429 . . . . . . . 8  |-  F/ x [_ w  /  x ]_ X  e.  B
31 csbeq1a 3089 . . . . . . . . 9  |-  ( x  =  w  ->  X  =  [_ w  /  x ]_ X )
3231eleq1d 2349 . . . . . . . 8  |-  ( x  =  w  ->  ( X  e.  B  <->  [_ w  /  x ]_ X  e.  B
) )
3330, 32rspc 2878 . . . . . . 7  |-  ( w  e.  A  ->  ( A. x  e.  A  X  e.  B  ->  [_ w  /  x ]_ X  e.  B )
)
3433com12 27 . . . . . 6  |-  ( A. x  e.  A  X  e.  B  ->  ( w  e.  A  ->  [_ w  /  x ]_ X  e.  B ) )
35 nfcsb1v 3113 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ X
3635nfel1 2429 . . . . . . . 8  |-  F/ x [_ z  /  x ]_ X  e.  B
37 csbeq1a 3089 . . . . . . . . 9  |-  ( x  =  z  ->  X  =  [_ z  /  x ]_ X )
3837eleq1d 2349 . . . . . . . 8  |-  ( x  =  z  ->  ( X  e.  B  <->  [_ z  /  x ]_ X  e.  B
) )
3936, 38rspc 2878 . . . . . . 7  |-  ( z  e.  A  ->  ( A. x  e.  A  X  e.  B  ->  [_ z  /  x ]_ X  e.  B )
)
4039com12 27 . . . . . 6  |-  ( A. x  e.  A  X  e.  B  ->  ( z  e.  A  ->  [_ z  /  x ]_ X  e.  B ) )
4128, 34, 403anim123d 1259 . . . . 5  |-  ( A. x  e.  A  X  e.  B  ->  ( ( v  e.  A  /\  w  e.  A  /\  z  e.  A )  ->  ( [_ v  /  x ]_ X  e.  B  /\  [_ w  /  x ]_ X  e.  B  /\  [_ z  /  x ]_ X  e.  B
) ) )
4241imp 418 . . . 4  |-  ( ( A. x  e.  A  X  e.  B  /\  ( v  e.  A  /\  w  e.  A  /\  z  e.  A
) )  ->  ( [_ v  /  x ]_ X  e.  B  /\  [_ w  /  x ]_ X  e.  B  /\  [_ z  /  x ]_ X  e.  B
) )
4342adantll 694 . . 3  |-  ( ( ( R  Po  B  /\  A. x  e.  A  X  e.  B )  /\  ( v  e.  A  /\  w  e.  A  /\  z  e.  A
) )  ->  ( [_ v  /  x ]_ X  e.  B  /\  [_ w  /  x ]_ X  e.  B  /\  [_ z  /  x ]_ X  e.  B
) )
44 potr 4326 . . . . 5  |-  ( ( R  Po  B  /\  ( [_ v  /  x ]_ X  e.  B  /\  [_ w  /  x ]_ X  e.  B  /\  [_ z  /  x ]_ X  e.  B
) )  ->  (
( [_ v  /  x ]_ X R [_ w  /  x ]_ X  /\  [_ w  /  x ]_ X R [_ z  /  x ]_ X )  ->  [_ v  /  x ]_ X R [_ z  /  x ]_ X ) )
45 df-br 4024 . . . . . . 7  |-  ( v S w  <->  <. v ,  w >.  e.  S
)
469eleq2i 2347 . . . . . . 7  |-  ( <.
v ,  w >.  e.  S  <->  <. v ,  w >.  e.  { <. x ,  y >.  |  X R Y } )
47 nfv 1605 . . . . . . . 8  |-  F/ y
[_ v  /  x ]_ X R [_ w  /  x ]_ X
48 vex 2791 . . . . . . . 8  |-  w  e. 
_V
49 csbeq1 3084 . . . . . . . . . 10  |-  ( y  =  w  ->  [_ y  /  x ]_ X  = 
[_ w  /  x ]_ X )
5019, 49syl5eqr 2329 . . . . . . . . 9  |-  ( y  =  w  ->  Y  =  [_ w  /  x ]_ X )
5150breq2d 4035 . . . . . . . 8  |-  ( y  =  w  ->  ( [_ v  /  x ]_ X R Y  <->  [_ v  /  x ]_ X R [_ w  /  x ]_ X
) )
5213, 47, 15, 48, 16, 51opelopabf 4289 . . . . . . 7  |-  ( <.
v ,  w >.  e. 
{ <. x ,  y
>.  |  X R Y }  <->  [_ v  /  x ]_ X R [_ w  /  x ]_ X )
5345, 46, 523bitri 262 . . . . . 6  |-  ( v S w  <->  [_ v  /  x ]_ X R [_ w  /  x ]_ X
)
54 df-br 4024 . . . . . . 7  |-  ( w S z  <->  <. w ,  z >.  e.  S
)
559eleq2i 2347 . . . . . . 7  |-  ( <.
w ,  z >.  e.  S  <->  <. w ,  z
>.  e.  { <. x ,  y >.  |  X R Y } )
5629, 11, 12nfbr 4067 . . . . . . . 8  |-  F/ x [_ w  /  x ]_ X R Y
57 nfv 1605 . . . . . . . 8  |-  F/ y
[_ w  /  x ]_ X R [_ z  /  x ]_ X
58 vex 2791 . . . . . . . 8  |-  z  e. 
_V
5931breq1d 4033 . . . . . . . 8  |-  ( x  =  w  ->  ( X R Y  <->  [_ w  /  x ]_ X R Y ) )
60 csbeq1 3084 . . . . . . . . . 10  |-  ( y  =  z  ->  [_ y  /  x ]_ X  = 
[_ z  /  x ]_ X )
6119, 60syl5eqr 2329 . . . . . . . . 9  |-  ( y  =  z  ->  Y  =  [_ z  /  x ]_ X )
6261breq2d 4035 . . . . . . . 8  |-  ( y  =  z  ->  ( [_ w  /  x ]_ X R Y  <->  [_ w  /  x ]_ X R [_ z  /  x ]_ X
) )
6356, 57, 48, 58, 59, 62opelopabf 4289 . . . . . . 7  |-  ( <.
w ,  z >.  e.  { <. x ,  y
>.  |  X R Y }  <->  [_ w  /  x ]_ X R [_ z  /  x ]_ X )
6454, 55, 633bitri 262 . . . . . 6  |-  ( w S z  <->  [_ w  /  x ]_ X R [_ z  /  x ]_ X
)
6553, 64anbi12i 678 . . . . 5  |-  ( ( v S w  /\  w S z )  <->  ( [_ v  /  x ]_ X R [_ w  /  x ]_ X  /\  [_ w  /  x ]_ X R
[_ z  /  x ]_ X ) )
66 df-br 4024 . . . . . 6  |-  ( v S z  <->  <. v ,  z >.  e.  S
)
679eleq2i 2347 . . . . . 6  |-  ( <.
v ,  z >.  e.  S  <->  <. v ,  z
>.  e.  { <. x ,  y >.  |  X R Y } )
68 nfv 1605 . . . . . . 7  |-  F/ y
[_ v  /  x ]_ X R [_ z  /  x ]_ X
6961breq2d 4035 . . . . . . 7  |-  ( y  =  z  ->  ( [_ v  /  x ]_ X R Y  <->  [_ v  /  x ]_ X R [_ z  /  x ]_ X
) )
7013, 68, 15, 58, 16, 69opelopabf 4289 . . . . . 6  |-  ( <.
v ,  z >.  e.  { <. x ,  y
>.  |  X R Y }  <->  [_ v  /  x ]_ X R [_ z  /  x ]_ X )
7166, 67, 703bitri 262 . . . . 5  |-  ( v S z  <->  [_ v  /  x ]_ X R [_ z  /  x ]_ X
)
7244, 65, 713imtr4g 261 . . . 4  |-  ( ( R  Po  B  /\  ( [_ v  /  x ]_ X  e.  B  /\  [_ w  /  x ]_ X  e.  B  /\  [_ z  /  x ]_ X  e.  B
) )  ->  (
( v S w  /\  w S z )  ->  v S
z ) )
7372adantlr 695 . . 3  |-  ( ( ( R  Po  B  /\  A. x  e.  A  X  e.  B )  /\  ( [_ v  /  x ]_ X  e.  B  /\  [_ w  /  x ]_ X  e.  B  /\  [_ z  /  x ]_ X  e.  B
) )  ->  (
( v S w  /\  w S z )  ->  v S
z ) )
7443, 73syldan 456 . 2  |-  ( ( ( R  Po  B  /\  A. x  e.  A  X  e.  B )  /\  ( v  e.  A  /\  w  e.  A  /\  z  e.  A
) )  ->  (
( v S w  /\  w S z )  ->  v S
z ) )
7527, 74ispod 4322 1  |-  ( ( R  Po  B  /\  A. x  e.  A  X  e.  B )  ->  S  Po  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   [_csb 3081   <.cop 3643   class class class wbr 4023   {copab 4076    Po wpo 4312
This theorem is referenced by:  pofunOLD  26428
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-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-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-po 4314
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