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Theorem pofun 4768
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 3414 . . . . . . 7  |-  F/_ x [_ v  /  x ]_ X
21nfel1 2632 . . . . . 6  |-  F/ x [_ v  /  x ]_ X  e.  B
3 csbeq1a 3407 . . . . . . 7  |-  ( x  =  v  ->  X  =  [_ v  /  x ]_ X )
43eleq1d 2523 . . . . . 6  |-  ( x  =  v  ->  ( X  e.  B  <->  [_ v  /  x ]_ X  e.  B
) )
52, 4rspc 3173 . . . . 5  |-  ( v  e.  A  ->  ( A. x  e.  A  X  e.  B  ->  [_ v  /  x ]_ X  e.  B )
)
65impcom 430 . . . 4  |-  ( ( A. x  e.  A  X  e.  B  /\  v  e.  A )  ->  [_ v  /  x ]_ X  e.  B
)
7 poirr 4763 . . . . 5  |-  ( ( R  Po  B  /\  [_ v  /  x ]_ X  e.  B )  ->  -.  [_ v  /  x ]_ X R [_ v  /  x ]_ X
)
8 df-br 4404 . . . . . 6  |-  ( v S v  <->  <. v ,  v >.  e.  S
)
9 pofun.1 . . . . . . 7  |-  S  =  { <. x ,  y
>.  |  X R Y }
109eleq2i 2532 . . . . . 6  |-  ( <.
v ,  v >.  e.  S  <->  <. v ,  v
>.  e.  { <. x ,  y >.  |  X R Y } )
11 nfcv 2616 . . . . . . . 8  |-  F/_ x R
12 nfcv 2616 . . . . . . . 8  |-  F/_ x Y
131, 11, 12nfbr 4447 . . . . . . 7  |-  F/ x [_ v  /  x ]_ X R Y
14 nfv 1674 . . . . . . 7  |-  F/ y
[_ v  /  x ]_ X R [_ v  /  x ]_ X
15 vex 3081 . . . . . . 7  |-  v  e. 
_V
163breq1d 4413 . . . . . . 7  |-  ( x  =  v  ->  ( X R Y  <->  [_ v  /  x ]_ X R Y ) )
17 vex 3081 . . . . . . . . . 10  |-  y  e. 
_V
18 pofun.2 . . . . . . . . . 10  |-  ( x  =  y  ->  X  =  Y )
1917, 12, 18csbief 3423 . . . . . . . . 9  |-  [_ y  /  x ]_ X  =  Y
20 csbeq1 3401 . . . . . . . . 9  |-  ( y  =  v  ->  [_ y  /  x ]_ X  = 
[_ v  /  x ]_ X )
2119, 20syl5eqr 2509 . . . . . . . 8  |-  ( y  =  v  ->  Y  =  [_ v  /  x ]_ X )
2221breq2d 4415 . . . . . . 7  |-  ( y  =  v  ->  ( [_ v  /  x ]_ X R Y  <->  [_ v  /  x ]_ X R [_ v  /  x ]_ X
) )
2313, 14, 15, 15, 16, 22opelopabf 4724 . . . . . 6  |-  ( <.
v ,  v >.  e.  { <. x ,  y
>.  |  X R Y }  <->  [_ v  /  x ]_ X R [_ v  /  x ]_ X )
248, 10, 233bitri 271 . . . . 5  |-  ( v S v  <->  [_ v  /  x ]_ X R [_ v  /  x ]_ X
)
257, 24sylnibr 305 . . . 4  |-  ( ( R  Po  B  /\  [_ v  /  x ]_ X  e.  B )  ->  -.  v S v )
266, 25sylan2 474 . . 3  |-  ( ( R  Po  B  /\  ( A. x  e.  A  X  e.  B  /\  v  e.  A )
)  ->  -.  v S v )
2726anassrs 648 . 2  |-  ( ( ( R  Po  B  /\  A. x  e.  A  X  e.  B )  /\  v  e.  A
)  ->  -.  v S v )
285com12 31 . . . . . 6  |-  ( A. x  e.  A  X  e.  B  ->  ( v  e.  A  ->  [_ v  /  x ]_ X  e.  B ) )
29 nfcsb1v 3414 . . . . . . . . 9  |-  F/_ x [_ w  /  x ]_ X
3029nfel1 2632 . . . . . . . 8  |-  F/ x [_ w  /  x ]_ X  e.  B
31 csbeq1a 3407 . . . . . . . . 9  |-  ( x  =  w  ->  X  =  [_ w  /  x ]_ X )
3231eleq1d 2523 . . . . . . . 8  |-  ( x  =  w  ->  ( X  e.  B  <->  [_ w  /  x ]_ X  e.  B
) )
3330, 32rspc 3173 . . . . . . 7  |-  ( w  e.  A  ->  ( A. x  e.  A  X  e.  B  ->  [_ w  /  x ]_ X  e.  B )
)
3433com12 31 . . . . . 6  |-  ( A. x  e.  A  X  e.  B  ->  ( w  e.  A  ->  [_ w  /  x ]_ X  e.  B ) )
35 nfcsb1v 3414 . . . . . . . . 9  |-  F/_ x [_ z  /  x ]_ X
3635nfel1 2632 . . . . . . . 8  |-  F/ x [_ z  /  x ]_ X  e.  B
37 csbeq1a 3407 . . . . . . . . 9  |-  ( x  =  z  ->  X  =  [_ z  /  x ]_ X )
3837eleq1d 2523 . . . . . . . 8  |-  ( x  =  z  ->  ( X  e.  B  <->  [_ z  /  x ]_ X  e.  B
) )
3936, 38rspc 3173 . . . . . . 7  |-  ( z  e.  A  ->  ( A. x  e.  A  X  e.  B  ->  [_ z  /  x ]_ X  e.  B )
)
4039com12 31 . . . . . 6  |-  ( A. x  e.  A  X  e.  B  ->  ( z  e.  A  ->  [_ z  /  x ]_ X  e.  B ) )
4128, 34, 403anim123d 1297 . . . . 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 429 . . . 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 713 . . 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 4764 . . . . 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 4404 . . . . . . 7  |-  ( v S w  <->  <. v ,  w >.  e.  S
)
469eleq2i 2532 . . . . . . 7  |-  ( <.
v ,  w >.  e.  S  <->  <. v ,  w >.  e.  { <. x ,  y >.  |  X R Y } )
47 nfv 1674 . . . . . . . 8  |-  F/ y
[_ v  /  x ]_ X R [_ w  /  x ]_ X
48 vex 3081 . . . . . . . 8  |-  w  e. 
_V
49 csbeq1 3401 . . . . . . . . . 10  |-  ( y  =  w  ->  [_ y  /  x ]_ X  = 
[_ w  /  x ]_ X )
5019, 49syl5eqr 2509 . . . . . . . . 9  |-  ( y  =  w  ->  Y  =  [_ w  /  x ]_ X )
5150breq2d 4415 . . . . . . . 8  |-  ( y  =  w  ->  ( [_ v  /  x ]_ X R Y  <->  [_ v  /  x ]_ X R [_ w  /  x ]_ X
) )
5213, 47, 15, 48, 16, 51opelopabf 4724 . . . . . . 7  |-  ( <.
v ,  w >.  e. 
{ <. x ,  y
>.  |  X R Y }  <->  [_ v  /  x ]_ X R [_ w  /  x ]_ X )
5345, 46, 523bitri 271 . . . . . 6  |-  ( v S w  <->  [_ v  /  x ]_ X R [_ w  /  x ]_ X
)
54 df-br 4404 . . . . . . 7  |-  ( w S z  <->  <. w ,  z >.  e.  S
)
559eleq2i 2532 . . . . . . 7  |-  ( <.
w ,  z >.  e.  S  <->  <. w ,  z
>.  e.  { <. x ,  y >.  |  X R Y } )
5629, 11, 12nfbr 4447 . . . . . . . 8  |-  F/ x [_ w  /  x ]_ X R Y
57 nfv 1674 . . . . . . . 8  |-  F/ y
[_ w  /  x ]_ X R [_ z  /  x ]_ X
58 vex 3081 . . . . . . . 8  |-  z  e. 
_V
5931breq1d 4413 . . . . . . . 8  |-  ( x  =  w  ->  ( X R Y  <->  [_ w  /  x ]_ X R Y ) )
60 csbeq1 3401 . . . . . . . . . 10  |-  ( y  =  z  ->  [_ y  /  x ]_ X  = 
[_ z  /  x ]_ X )
6119, 60syl5eqr 2509 . . . . . . . . 9  |-  ( y  =  z  ->  Y  =  [_ z  /  x ]_ X )
6261breq2d 4415 . . . . . . . 8  |-  ( y  =  z  ->  ( [_ w  /  x ]_ X R Y  <->  [_ w  /  x ]_ X R [_ z  /  x ]_ X
) )
6356, 57, 48, 58, 59, 62opelopabf 4724 . . . . . . 7  |-  ( <.
w ,  z >.  e.  { <. x ,  y
>.  |  X R Y }  <->  [_ w  /  x ]_ X R [_ z  /  x ]_ X )
6454, 55, 633bitri 271 . . . . . 6  |-  ( w S z  <->  [_ w  /  x ]_ X R [_ z  /  x ]_ X
)
6553, 64anbi12i 697 . . . . 5  |-  ( ( v S w  /\  w S z )  <->  ( [_ v  /  x ]_ X R [_ w  /  x ]_ X  /\  [_ w  /  x ]_ X R
[_ z  /  x ]_ X ) )
66 df-br 4404 . . . . . 6  |-  ( v S z  <->  <. v ,  z >.  e.  S
)
679eleq2i 2532 . . . . . 6  |-  ( <.
v ,  z >.  e.  S  <->  <. v ,  z
>.  e.  { <. x ,  y >.  |  X R Y } )
68 nfv 1674 . . . . . . 7  |-  F/ y
[_ v  /  x ]_ X R [_ z  /  x ]_ X
6961breq2d 4415 . . . . . . 7  |-  ( y  =  z  ->  ( [_ v  /  x ]_ X R Y  <->  [_ v  /  x ]_ X R [_ z  /  x ]_ X
) )
7013, 68, 15, 58, 16, 69opelopabf 4724 . . . . . 6  |-  ( <.
v ,  z >.  e.  { <. x ,  y
>.  |  X R Y }  <->  [_ v  /  x ]_ X R [_ z  /  x ]_ X )
7166, 67, 703bitri 271 . . . . 5  |-  ( v S z  <->  [_ v  /  x ]_ X R [_ z  /  x ]_ X
)
7244, 65, 713imtr4g 270 . . . 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 714 . . 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 470 . 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 4760 1  |-  ( ( R  Po  B  /\  A. x  e.  A  X  e.  B )  ->  S  Po  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   A.wral 2799   [_csb 3398   <.cop 3994   class class class wbr 4403   {copab 4460    Po wpo 4750
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-po 4752
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator