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Theorem cusgrafilem2 23523
Description: Lemma 2 for cusgrafi 23525. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
Hypotheses
Ref Expression
cusgrafi.p  |-  P  =  { x  e.  ~P V  |  E. a  e.  V  ( a  =/=  N  /\  x  =  { a ,  N } ) }
cusgrafi.f  |-  F  =  ( x  e.  ( V  \  { N } )  |->  { x ,  N } )
Assertion
Ref Expression
cusgrafilem2  |-  ( ( V  e.  W  /\  N  e.  V )  ->  F : ( V 
\  { N }
)
-1-1-onto-> P )
Distinct variable groups:    N, a, x    V, a, x    x, P    W, a, x
Allowed substitution hints:    P( a)    F( x, a)

Proof of Theorem cusgrafilem2
Dummy variables  e 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifsn 4098 . . . . . . 7  |-  ( v  e.  ( V  \  { N } )  <->  ( v  e.  V  /\  v  =/=  N ) )
2 simpl 457 . . . . . . 7  |-  ( ( v  e.  V  /\  v  =/=  N )  -> 
v  e.  V )
31, 2sylbi 195 . . . . . 6  |-  ( v  e.  ( V  \  { N } )  -> 
v  e.  V )
4 simpr 461 . . . . . 6  |-  ( ( V  e.  W  /\  N  e.  V )  ->  N  e.  V )
5 prelpwi 4637 . . . . . 6  |-  ( ( v  e.  V  /\  N  e.  V )  ->  { v ,  N }  e.  ~P V
)
63, 4, 5syl2anr 478 . . . . 5  |-  ( ( ( V  e.  W  /\  N  e.  V
)  /\  v  e.  ( V  \  { N } ) )  ->  { v ,  N }  e.  ~P V
)
71biimpi 194 . . . . . . 7  |-  ( v  e.  ( V  \  { N } )  -> 
( v  e.  V  /\  v  =/=  N
) )
87adantl 466 . . . . . 6  |-  ( ( ( V  e.  W  /\  N  e.  V
)  /\  v  e.  ( V  \  { N } ) )  -> 
( v  e.  V  /\  v  =/=  N
) )
9 simpr 461 . . . . . . . . 9  |-  ( ( v  e.  V  /\  v  =/=  N )  -> 
v  =/=  N )
101, 9sylbi 195 . . . . . . . 8  |-  ( v  e.  ( V  \  { N } )  -> 
v  =/=  N )
1110adantl 466 . . . . . . 7  |-  ( ( ( V  e.  W  /\  N  e.  V
)  /\  v  e.  ( V  \  { N } ) )  -> 
v  =/=  N )
12 eqidd 2452 . . . . . . 7  |-  ( ( ( V  e.  W  /\  N  e.  V
)  /\  v  e.  ( V  \  { N } ) )  ->  { v ,  N }  =  { v ,  N } )
1311, 12jca 532 . . . . . 6  |-  ( ( ( V  e.  W  /\  N  e.  V
)  /\  v  e.  ( V  \  { N } ) )  -> 
( v  =/=  N  /\  { v ,  N }  =  { v ,  N } ) )
14 neeq1 2729 . . . . . . . . 9  |-  ( a  =  v  ->  (
a  =/=  N  <->  v  =/=  N ) )
15 preq1 4052 . . . . . . . . . 10  |-  ( a  =  v  ->  { a ,  N }  =  { v ,  N } )
1615eqeq2d 2465 . . . . . . . . 9  |-  ( a  =  v  ->  ( { v ,  N }  =  { a ,  N }  <->  { v ,  N }  =  {
v ,  N }
) )
1714, 16anbi12d 710 . . . . . . . 8  |-  ( a  =  v  ->  (
( a  =/=  N  /\  { v ,  N }  =  { a ,  N } )  <->  ( v  =/=  N  /\  { v ,  N }  =  { v ,  N } ) ) )
1817adantl 466 . . . . . . 7  |-  ( ( ( v  e.  V  /\  v  =/=  N
)  /\  a  =  v )  ->  (
( a  =/=  N  /\  { v ,  N }  =  { a ,  N } )  <->  ( v  =/=  N  /\  { v ,  N }  =  { v ,  N } ) ) )
192, 18rspcedv 3173 . . . . . 6  |-  ( ( v  e.  V  /\  v  =/=  N )  -> 
( ( v  =/= 
N  /\  { v ,  N }  =  {
v ,  N }
)  ->  E. a  e.  V  ( a  =/=  N  /\  { v ,  N }  =  { a ,  N } ) ) )
208, 13, 19sylc 60 . . . . 5  |-  ( ( ( V  e.  W  /\  N  e.  V
)  /\  v  e.  ( V  \  { N } ) )  ->  E. a  e.  V  ( a  =/=  N  /\  { v ,  N }  =  { a ,  N } ) )
21 eqeq1 2455 . . . . . . . 8  |-  ( x  =  { v ,  N }  ->  (
x  =  { a ,  N }  <->  { v ,  N }  =  {
a ,  N }
) )
2221anbi2d 703 . . . . . . 7  |-  ( x  =  { v ,  N }  ->  (
( a  =/=  N  /\  x  =  {
a ,  N }
)  <->  ( a  =/= 
N  /\  { v ,  N }  =  {
a ,  N }
) ) )
2322rexbidv 2844 . . . . . 6  |-  ( x  =  { v ,  N }  ->  ( E. a  e.  V  ( a  =/=  N  /\  x  =  {
a ,  N }
)  <->  E. a  e.  V  ( a  =/=  N  /\  { v ,  N }  =  { a ,  N } ) ) )
24 cusgrafi.p . . . . . 6  |-  P  =  { x  e.  ~P V  |  E. a  e.  V  ( a  =/=  N  /\  x  =  { a ,  N } ) }
2523, 24elrab2 3216 . . . . 5  |-  ( { v ,  N }  e.  P  <->  ( { v ,  N }  e.  ~P V  /\  E. a  e.  V  ( a  =/=  N  /\  { v ,  N }  =  { a ,  N } ) ) )
266, 20, 25sylanbrc 664 . . . 4  |-  ( ( ( V  e.  W  /\  N  e.  V
)  /\  v  e.  ( V  \  { N } ) )  ->  { v ,  N }  e.  P )
2726ralrimiva 2822 . . 3  |-  ( ( V  e.  W  /\  N  e.  V )  ->  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  P )
28 preq1 4052 . . . . 5  |-  ( x  =  v  ->  { x ,  N }  =  {
v ,  N }
)
2928eleq1d 2520 . . . 4  |-  ( x  =  v  ->  ( { x ,  N }  e.  P  <->  { v ,  N }  e.  P
) )
3029cbvralv 3043 . . 3  |-  ( A. x  e.  ( V  \  { N } ) { x ,  N }  e.  P  <->  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  P )
3127, 30sylibr 212 . 2  |-  ( ( V  e.  W  /\  N  e.  V )  ->  A. x  e.  ( V  \  { N } ) { x ,  N }  e.  P
)
32 simpl 457 . . . . . . . . . . 11  |-  ( ( a  =/=  N  /\  e  =  { a ,  N } )  -> 
a  =/=  N )
3332anim2i 569 . . . . . . . . . 10  |-  ( ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) )  ->  (
a  e.  V  /\  a  =/=  N ) )
3433adantl 466 . . . . . . . . 9  |-  ( ( ( ( V  e.  W  /\  N  e.  V )  /\  e  e.  ~P V )  /\  ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) ) )  -> 
( a  e.  V  /\  a  =/=  N
) )
35 eldifsn 4098 . . . . . . . . 9  |-  ( a  e.  ( V  \  { N } )  <->  ( a  e.  V  /\  a  =/=  N ) )
3634, 35sylibr 212 . . . . . . . 8  |-  ( ( ( ( V  e.  W  /\  N  e.  V )  /\  e  e.  ~P V )  /\  ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) ) )  -> 
a  e.  ( V 
\  { N }
) )
37 eqeq1 2455 . . . . . . . . . . . . . 14  |-  ( e  =  { a ,  N }  ->  (
e  =  { x ,  N }  <->  { a ,  N }  =  {
x ,  N }
) )
3837adantl 466 . . . . . . . . . . . . 13  |-  ( ( a  =/=  N  /\  e  =  { a ,  N } )  -> 
( e  =  {
x ,  N }  <->  { a ,  N }  =  { x ,  N } ) )
3938ad2antlr 726 . . . . . . . . . . . 12  |-  ( ( ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) )  /\  x  e.  ( V  \  { N } ) )  -> 
( e  =  {
x ,  N }  <->  { a ,  N }  =  { x ,  N } ) )
40 vex 3071 . . . . . . . . . . . . . 14  |-  a  e. 
_V
41 vex 3071 . . . . . . . . . . . . . 14  |-  x  e. 
_V
4240, 41preqr1 4144 . . . . . . . . . . . . 13  |-  ( { a ,  N }  =  { x ,  N }  ->  a  =  x )
4342eqcomd 2459 . . . . . . . . . . . 12  |-  ( { a ,  N }  =  { x ,  N }  ->  x  =  a )
4439, 43syl6bi 228 . . . . . . . . . . 11  |-  ( ( ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) )  /\  x  e.  ( V  \  { N } ) )  -> 
( e  =  {
x ,  N }  ->  x  =  a ) )
4544adantll 713 . . . . . . . . . 10  |-  ( ( ( ( ( V  e.  W  /\  N  e.  V )  /\  e  e.  ~P V )  /\  ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) ) )  /\  x  e.  ( V  \  { N } ) )  ->  ( e  =  { x ,  N }  ->  x  =  a ) )
46 preq1 4052 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  { a ,  N }  =  { x ,  N } )
4746equcoms 1735 . . . . . . . . . . . . . . 15  |-  ( x  =  a  ->  { a ,  N }  =  { x ,  N } )
4847eqeq2d 2465 . . . . . . . . . . . . . 14  |-  ( x  =  a  ->  (
e  =  { a ,  N }  <->  e  =  { x ,  N } ) )
4948biimpcd 224 . . . . . . . . . . . . 13  |-  ( e  =  { a ,  N }  ->  (
x  =  a  -> 
e  =  { x ,  N } ) )
5049adantl 466 . . . . . . . . . . . 12  |-  ( ( a  =/=  N  /\  e  =  { a ,  N } )  -> 
( x  =  a  ->  e  =  {
x ,  N }
) )
5150adantl 466 . . . . . . . . . . 11  |-  ( ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) )  ->  (
x  =  a  -> 
e  =  { x ,  N } ) )
5251ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ( ( V  e.  W  /\  N  e.  V )  /\  e  e.  ~P V )  /\  ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) ) )  /\  x  e.  ( V  \  { N } ) )  ->  ( x  =  a  ->  e  =  { x ,  N } ) )
5345, 52impbid 191 . . . . . . . . 9  |-  ( ( ( ( ( V  e.  W  /\  N  e.  V )  /\  e  e.  ~P V )  /\  ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) ) )  /\  x  e.  ( V  \  { N } ) )  ->  ( e  =  { x ,  N } 
<->  x  =  a ) )
5453ralrimiva 2822 . . . . . . . 8  |-  ( ( ( ( V  e.  W  /\  N  e.  V )  /\  e  e.  ~P V )  /\  ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) ) )  ->  A. x  e.  ( V  \  { N }
) ( e  =  { x ,  N } 
<->  x  =  a ) )
5536, 54jca 532 . . . . . . 7  |-  ( ( ( ( V  e.  W  /\  N  e.  V )  /\  e  e.  ~P V )  /\  ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) ) )  -> 
( a  e.  ( V  \  { N } )  /\  A. x  e.  ( V  \  { N } ) ( e  =  {
x ,  N }  <->  x  =  a ) ) )
5655ex 434 . . . . . 6  |-  ( ( ( V  e.  W  /\  N  e.  V
)  /\  e  e.  ~P V )  ->  (
( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) )  ->  (
a  e.  ( V 
\  { N }
)  /\  A. x  e.  ( V  \  { N } ) ( e  =  { x ,  N }  <->  x  =  a ) ) ) )
5756reximdv2 2921 . . . . 5  |-  ( ( ( V  e.  W  /\  N  e.  V
)  /\  e  e.  ~P V )  ->  ( E. a  e.  V  ( a  =/=  N  /\  e  =  {
a ,  N }
)  ->  E. a  e.  ( V  \  { N } ) A. x  e.  ( V  \  { N } ) ( e  =  { x ,  N }  <->  x  =  a ) ) )
5857expimpd 603 . . . 4  |-  ( ( V  e.  W  /\  N  e.  V )  ->  ( ( e  e. 
~P V  /\  E. a  e.  V  (
a  =/=  N  /\  e  =  { a ,  N } ) )  ->  E. a  e.  ( V  \  { N } ) A. x  e.  ( V  \  { N } ) ( e  =  { x ,  N }  <->  x  =  a ) ) )
59 eqeq1 2455 . . . . . . 7  |-  ( x  =  e  ->  (
x  =  { a ,  N }  <->  e  =  { a ,  N } ) )
6059anbi2d 703 . . . . . 6  |-  ( x  =  e  ->  (
( a  =/=  N  /\  x  =  {
a ,  N }
)  <->  ( a  =/= 
N  /\  e  =  { a ,  N } ) ) )
6160rexbidv 2844 . . . . 5  |-  ( x  =  e  ->  ( E. a  e.  V  ( a  =/=  N  /\  x  =  {
a ,  N }
)  <->  E. a  e.  V  ( a  =/=  N  /\  e  =  {
a ,  N }
) ) )
6261, 24elrab2 3216 . . . 4  |-  ( e  e.  P  <->  ( e  e.  ~P V  /\  E. a  e.  V  (
a  =/=  N  /\  e  =  { a ,  N } ) ) )
63 reu6 3245 . . . 4  |-  ( E! x  e.  ( V 
\  { N }
) e  =  {
x ,  N }  <->  E. a  e.  ( V 
\  { N }
) A. x  e.  ( V  \  { N } ) ( e  =  { x ,  N }  <->  x  =  a ) )
6458, 62, 633imtr4g 270 . . 3  |-  ( ( V  e.  W  /\  N  e.  V )  ->  ( e  e.  P  ->  E! x  e.  ( V  \  { N } ) e  =  { x ,  N } ) )
6564ralrimiv 2820 . 2  |-  ( ( V  e.  W  /\  N  e.  V )  ->  A. e  e.  P  E! x  e.  ( V  \  { N }
) e  =  {
x ,  N }
)
66 cusgrafi.f . . 3  |-  F  =  ( x  e.  ( V  \  { N } )  |->  { x ,  N } )
6766f1ompt 5964 . 2  |-  ( F : ( V  \  { N } ) -1-1-onto-> P  <->  ( A. x  e.  ( V  \  { N } ) { x ,  N }  e.  P  /\  A. e  e.  P  E! x  e.  ( V  \  { N } ) e  =  { x ,  N } ) )
6831, 65, 67sylanbrc 664 1  |-  ( ( V  e.  W  /\  N  e.  V )  ->  F : ( V 
\  { N }
)
-1-1-onto-> P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   E.wrex 2796   E!wreu 2797   {crab 2799    \ cdif 3423   ~Pcpw 3958   {csn 3975   {cpr 3977    |-> cmpt 4448   -1-1-onto->wf1o 5515
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-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-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524
This theorem is referenced by:  cusgrafilem3  23524
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