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Theorem wlkntrllem3 25291
Description: Lemma 3 for wlkntrl 25292: F is not injective. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
Hypotheses
Ref Expression
wlkntrl.v  |-  V  =  { x ,  y }
wlkntrl.e  |-  E  =  { <. 0 ,  {
x ,  y }
>. }
wlkntrl.f  |-  F  =  { <. 0 ,  0
>. ,  <. 1 ,  0 >. }
wlkntrl.p  |-  P  =  { <. 0 ,  x >. ,  <. 1 ,  y
>. ,  <. 2 ,  x >. }
Assertion
Ref Expression
wlkntrllem3  |-  -.  Fun  `' F
Distinct variable group:    x, F, y
Allowed substitution hints:    P( x, y)    E( x, y)    V( x, y)

Proof of Theorem wlkntrllem3
StepHypRef Expression
1 ax-1ne0 9608 . . 3  |-  1  =/=  0
21neii 2626 . 2  |-  -.  1  =  0
3 0ne1 10677 . . . . 5  |-  0  =/=  1
4 c0ex 9637 . . . . . 6  |-  0  e.  _V
5 1ex 9638 . . . . . 6  |-  1  e.  _V
64, 5, 4, 4fpr 6072 . . . . 5  |-  ( 0  =/=  1  ->  { <. 0 ,  0 >. , 
<. 1 ,  0
>. } : { 0 ,  1 } --> { 0 ,  0 } )
73, 6ax-mp 5 . . . 4  |-  { <. 0 ,  0 >. , 
<. 1 ,  0
>. } : { 0 ,  1 } --> { 0 ,  0 }
8 wlkntrl.f . . . . . 6  |-  F  =  { <. 0 ,  0
>. ,  <. 1 ,  0 >. }
98eqcomi 2460 . . . . 5  |-  { <. 0 ,  0 >. , 
<. 1 ,  0
>. }  =  F
109feq1i 5720 . . . 4  |-  ( {
<. 0 ,  0
>. ,  <. 1 ,  0 >. } : {
0 ,  1 } --> { 0 ,  0 }  <->  F : { 0 ,  1 } --> { 0 ,  0 } )
117, 10mpbi 212 . . 3  |-  F : { 0 ,  1 } --> { 0 ,  0 }
12 df-f1 5587 . . . 4  |-  ( F : { 0 ,  1 } -1-1-> { 0 ,  0 }  <->  ( F : { 0 ,  1 } --> { 0 ,  0 }  /\  Fun  `' F ) )
13 dff13 6159 . . . . 5  |-  ( F : { 0 ,  1 } -1-1-> { 0 ,  0 }  <->  ( F : { 0 ,  1 } --> { 0 ,  0 }  /\  A. x  e.  { 0 ,  1 } A. y  e.  { 0 ,  1 }  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )
) )
14 fveq2 5865 . . . . . . . . . . 11  |-  ( x  =  0  ->  ( F `  x )  =  ( F ` 
0 ) )
1514eqeq1d 2453 . . . . . . . . . 10  |-  ( x  =  0  ->  (
( F `  x
)  =  ( F `
 y )  <->  ( F `  0 )  =  ( F `  y
) ) )
16 eqeq1 2455 . . . . . . . . . 10  |-  ( x  =  0  ->  (
x  =  y  <->  0  =  y ) )
1715, 16imbi12d 322 . . . . . . . . 9  |-  ( x  =  0  ->  (
( ( F `  x )  =  ( F `  y )  ->  x  =  y )  <->  ( ( F `
 0 )  =  ( F `  y
)  ->  0  =  y ) ) )
1817ralbidv 2827 . . . . . . . 8  |-  ( x  =  0  ->  ( A. y  e.  { 0 ,  1 }  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )  <->  A. y  e.  { 0 ,  1 }  (
( F `  0
)  =  ( F `
 y )  -> 
0  =  y ) ) )
19 fveq2 5865 . . . . . . . . . . 11  |-  ( x  =  1  ->  ( F `  x )  =  ( F ` 
1 ) )
2019eqeq1d 2453 . . . . . . . . . 10  |-  ( x  =  1  ->  (
( F `  x
)  =  ( F `
 y )  <->  ( F `  1 )  =  ( F `  y
) ) )
21 eqeq1 2455 . . . . . . . . . 10  |-  ( x  =  1  ->  (
x  =  y  <->  1  =  y ) )
2220, 21imbi12d 322 . . . . . . . . 9  |-  ( x  =  1  ->  (
( ( F `  x )  =  ( F `  y )  ->  x  =  y )  <->  ( ( F `
 1 )  =  ( F `  y
)  ->  1  =  y ) ) )
2322ralbidv 2827 . . . . . . . 8  |-  ( x  =  1  ->  ( A. y  e.  { 0 ,  1 }  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )  <->  A. y  e.  { 0 ,  1 }  (
( F `  1
)  =  ( F `
 y )  -> 
1  =  y ) ) )
244, 5, 18, 23ralpr 4025 . . . . . . 7  |-  ( A. x  e.  { 0 ,  1 } A. y  e.  { 0 ,  1 }  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )  <->  ( A. y  e.  {
0 ,  1 }  ( ( F ` 
0 )  =  ( F `  y )  ->  0  =  y )  /\  A. y  e.  { 0 ,  1 }  ( ( F `
 1 )  =  ( F `  y
)  ->  1  =  y ) ) )
25 fveq2 5865 . . . . . . . . . . 11  |-  ( y  =  0  ->  ( F `  y )  =  ( F ` 
0 ) )
2625eqeq2d 2461 . . . . . . . . . 10  |-  ( y  =  0  ->  (
( F `  0
)  =  ( F `
 y )  <->  ( F `  0 )  =  ( F `  0
) ) )
27 eqeq2 2462 . . . . . . . . . 10  |-  ( y  =  0  ->  (
0  =  y  <->  0  = 
0 ) )
2826, 27imbi12d 322 . . . . . . . . 9  |-  ( y  =  0  ->  (
( ( F ` 
0 )  =  ( F `  y )  ->  0  =  y )  <->  ( ( F `
 0 )  =  ( F `  0
)  ->  0  = 
0 ) ) )
29 fveq2 5865 . . . . . . . . . . 11  |-  ( y  =  1  ->  ( F `  y )  =  ( F ` 
1 ) )
3029eqeq2d 2461 . . . . . . . . . 10  |-  ( y  =  1  ->  (
( F `  0
)  =  ( F `
 y )  <->  ( F `  0 )  =  ( F `  1
) ) )
31 eqeq2 2462 . . . . . . . . . 10  |-  ( y  =  1  ->  (
0  =  y  <->  0  = 
1 ) )
3230, 31imbi12d 322 . . . . . . . . 9  |-  ( y  =  1  ->  (
( ( F ` 
0 )  =  ( F `  y )  ->  0  =  y )  <->  ( ( F `
 0 )  =  ( F `  1
)  ->  0  = 
1 ) ) )
334, 5, 28, 32ralpr 4025 . . . . . . . 8  |-  ( A. y  e.  { 0 ,  1 }  (
( F `  0
)  =  ( F `
 y )  -> 
0  =  y )  <-> 
( ( ( F `
 0 )  =  ( F `  0
)  ->  0  = 
0 )  /\  (
( F `  0
)  =  ( F `
 1 )  -> 
0  =  1 ) ) )
3425eqeq2d 2461 . . . . . . . . . 10  |-  ( y  =  0  ->  (
( F `  1
)  =  ( F `
 y )  <->  ( F `  1 )  =  ( F `  0
) ) )
35 eqeq2 2462 . . . . . . . . . 10  |-  ( y  =  0  ->  (
1  =  y  <->  1  = 
0 ) )
3634, 35imbi12d 322 . . . . . . . . 9  |-  ( y  =  0  ->  (
( ( F ` 
1 )  =  ( F `  y )  ->  1  =  y )  <->  ( ( F `
 1 )  =  ( F `  0
)  ->  1  = 
0 ) ) )
3729eqeq2d 2461 . . . . . . . . . 10  |-  ( y  =  1  ->  (
( F `  1
)  =  ( F `
 y )  <->  ( F `  1 )  =  ( F `  1
) ) )
38 eqeq2 2462 . . . . . . . . . 10  |-  ( y  =  1  ->  (
1  =  y  <->  1  = 
1 ) )
3937, 38imbi12d 322 . . . . . . . . 9  |-  ( y  =  1  ->  (
( ( F ` 
1 )  =  ( F `  y )  ->  1  =  y )  <->  ( ( F `
 1 )  =  ( F `  1
)  ->  1  = 
1 ) ) )
404, 5, 36, 39ralpr 4025 . . . . . . . 8  |-  ( A. y  e.  { 0 ,  1 }  (
( F `  1
)  =  ( F `
 y )  -> 
1  =  y )  <-> 
( ( ( F `
 1 )  =  ( F `  0
)  ->  1  = 
0 )  /\  (
( F `  1
)  =  ( F `
 1 )  -> 
1  =  1 ) ) )
418fveq1i 5866 . . . . . . . . . . . . 13  |-  ( F `
 1 )  =  ( { <. 0 ,  0 >. ,  <. 1 ,  0 >. } `
 1 )
425, 4fvpr2 6108 . . . . . . . . . . . . . 14  |-  ( 0  =/=  1  ->  ( { <. 0 ,  0
>. ,  <. 1 ,  0 >. } `  1
)  =  0 )
433, 42mp1i 13 . . . . . . . . . . . . 13  |-  ( -.  1  =  0  -> 
( { <. 0 ,  0 >. ,  <. 1 ,  0 >. } `
 1 )  =  0 )
4441, 43syl5eq 2497 . . . . . . . . . . . 12  |-  ( -.  1  =  0  -> 
( F `  1
)  =  0 )
458fveq1i 5866 . . . . . . . . . . . . 13  |-  ( F `
 0 )  =  ( { <. 0 ,  0 >. ,  <. 1 ,  0 >. } `
 0 )
464, 4fvpr1 6107 . . . . . . . . . . . . . 14  |-  ( 0  =/=  1  ->  ( { <. 0 ,  0
>. ,  <. 1 ,  0 >. } `  0
)  =  0 )
473, 46mp1i 13 . . . . . . . . . . . . 13  |-  ( -.  1  =  0  -> 
( { <. 0 ,  0 >. ,  <. 1 ,  0 >. } `
 0 )  =  0 )
4845, 47syl5req 2498 . . . . . . . . . . . 12  |-  ( -.  1  =  0  -> 
0  =  ( F `
 0 ) )
4944, 48eqtrd 2485 . . . . . . . . . . 11  |-  ( -.  1  =  0  -> 
( F `  1
)  =  ( F `
 0 ) )
5049con1i 133 . . . . . . . . . 10  |-  ( -.  ( F `  1
)  =  ( F `
 0 )  -> 
1  =  0 )
51 id 22 . . . . . . . . . 10  |-  ( 1  =  0  ->  1  =  0 )
5250, 51ja 165 . . . . . . . . 9  |-  ( ( ( F `  1
)  =  ( F `
 0 )  -> 
1  =  0 )  ->  1  =  0 )
5352ad2antrl 734 . . . . . . . 8  |-  ( ( ( ( ( F `
 0 )  =  ( F `  0
)  ->  0  = 
0 )  /\  (
( F `  0
)  =  ( F `
 1 )  -> 
0  =  1 ) )  /\  ( ( ( F `  1
)  =  ( F `
 0 )  -> 
1  =  0 )  /\  ( ( F `
 1 )  =  ( F `  1
)  ->  1  = 
1 ) ) )  ->  1  =  0 )
5433, 40, 53syl2anb 482 . . . . . . 7  |-  ( ( A. y  e.  {
0 ,  1 }  ( ( F ` 
0 )  =  ( F `  y )  ->  0  =  y )  /\  A. y  e.  { 0 ,  1 }  ( ( F `
 1 )  =  ( F `  y
)  ->  1  =  y ) )  -> 
1  =  0 )
5524, 54sylbi 199 . . . . . 6  |-  ( A. x  e.  { 0 ,  1 } A. y  e.  { 0 ,  1 }  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )  ->  1  =  0 )
5655adantl 468 . . . . 5  |-  ( ( F : { 0 ,  1 } --> { 0 ,  0 }  /\  A. x  e.  { 0 ,  1 } A. y  e.  { 0 ,  1 }  (
( F `  x
)  =  ( F `
 y )  ->  x  =  y )
)  ->  1  = 
0 )
5713, 56sylbi 199 . . . 4  |-  ( F : { 0 ,  1 } -1-1-> { 0 ,  0 }  ->  1  =  0 )
5812, 57sylbir 217 . . 3  |-  ( ( F : { 0 ,  1 } --> { 0 ,  0 }  /\  Fun  `' F )  ->  1  =  0 )
5911, 58mpan 676 . 2  |-  ( Fun  `' F  ->  1  =  0 )
602, 59mto 180 1  |-  -.  Fun  `' F
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    = wceq 1444    =/= wne 2622   A.wral 2737   {csn 3968   {cpr 3970   {ctp 3972   <.cop 3974   `'ccnv 4833   Fun wfun 5576   -->wf 5578   -1-1->wf1 5579   ` cfv 5582   0cc0 9539   1c1 9540   2c2 10659
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-mulcl 9601  ax-i2m1 9607  ax-1ne0 9608
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fv 5590
This theorem is referenced by:  wlkntrl  25292
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