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Theorem cusgrafilem2 21489
Description: Lemma 2 for cusgrafi 21491. (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 3927 . . . . . . 7  |-  ( v  e.  ( V  \  { N } )  <->  ( v  e.  V  /\  v  =/=  N ) )
2 simpl 444 . . . . . . 7  |-  ( ( v  e.  V  /\  v  =/=  N )  -> 
v  e.  V )
31, 2sylbi 188 . . . . . 6  |-  ( v  e.  ( V  \  { N } )  -> 
v  e.  V )
4 simpr 448 . . . . . 6  |-  ( ( V  e.  W  /\  N  e.  V )  ->  N  e.  V )
5 prelpwi 4411 . . . . . 6  |-  ( ( v  e.  V  /\  N  e.  V )  ->  { v ,  N }  e.  ~P V
)
63, 4, 5syl2anr 465 . . . . 5  |-  ( ( ( V  e.  W  /\  N  e.  V
)  /\  v  e.  ( V  \  { N } ) )  ->  { v ,  N }  e.  ~P V
)
71biimpi 187 . . . . . . 7  |-  ( v  e.  ( V  \  { N } )  -> 
( v  e.  V  /\  v  =/=  N
) )
87adantl 453 . . . . . 6  |-  ( ( ( V  e.  W  /\  N  e.  V
)  /\  v  e.  ( V  \  { N } ) )  -> 
( v  e.  V  /\  v  =/=  N
) )
9 simpr 448 . . . . . . . . 9  |-  ( ( v  e.  V  /\  v  =/=  N )  -> 
v  =/=  N )
101, 9sylbi 188 . . . . . . . 8  |-  ( v  e.  ( V  \  { N } )  -> 
v  =/=  N )
1110adantl 453 . . . . . . 7  |-  ( ( ( V  e.  W  /\  N  e.  V
)  /\  v  e.  ( V  \  { N } ) )  -> 
v  =/=  N )
12 eqidd 2437 . . . . . . 7  |-  ( ( ( V  e.  W  /\  N  e.  V
)  /\  v  e.  ( V  \  { N } ) )  ->  { v ,  N }  =  { v ,  N } )
1311, 12jca 519 . . . . . 6  |-  ( ( ( V  e.  W  /\  N  e.  V
)  /\  v  e.  ( V  \  { N } ) )  -> 
( v  =/=  N  /\  { v ,  N }  =  { v ,  N } ) )
14 neeq1 2609 . . . . . . . . 9  |-  ( a  =  v  ->  (
a  =/=  N  <->  v  =/=  N ) )
15 preq1 3883 . . . . . . . . . 10  |-  ( a  =  v  ->  { a ,  N }  =  { v ,  N } )
1615eqeq2d 2447 . . . . . . . . 9  |-  ( a  =  v  ->  ( { v ,  N }  =  { a ,  N }  <->  { v ,  N }  =  {
v ,  N }
) )
1714, 16anbi12d 692 . . . . . . . 8  |-  ( a  =  v  ->  (
( a  =/=  N  /\  { v ,  N }  =  { a ,  N } )  <->  ( v  =/=  N  /\  { v ,  N }  =  { v ,  N } ) ) )
1817adantl 453 . . . . . . 7  |-  ( ( ( v  e.  V  /\  v  =/=  N
)  /\  a  =  v )  ->  (
( a  =/=  N  /\  { v ,  N }  =  { a ,  N } )  <->  ( v  =/=  N  /\  { v ,  N }  =  { v ,  N } ) ) )
192, 18rspcedv 3056 . . . . . 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 58 . . . . 5  |-  ( ( ( V  e.  W  /\  N  e.  V
)  /\  v  e.  ( V  \  { N } ) )  ->  E. a  e.  V  ( a  =/=  N  /\  { v ,  N }  =  { a ,  N } ) )
21 eqeq1 2442 . . . . . . . 8  |-  ( x  =  { v ,  N }  ->  (
x  =  { a ,  N }  <->  { v ,  N }  =  {
a ,  N }
) )
2221anbi2d 685 . . . . . . 7  |-  ( x  =  { v ,  N }  ->  (
( a  =/=  N  /\  x  =  {
a ,  N }
)  <->  ( a  =/= 
N  /\  { v ,  N }  =  {
a ,  N }
) ) )
2322rexbidv 2726 . . . . . 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 3094 . . . . 5  |-  ( { v ,  N }  e.  P  <->  ( { v ,  N }  e.  ~P V  /\  E. a  e.  V  ( a  =/=  N  /\  { v ,  N }  =  { a ,  N } ) ) )
266, 20, 25sylanbrc 646 . . . 4  |-  ( ( ( V  e.  W  /\  N  e.  V
)  /\  v  e.  ( V  \  { N } ) )  ->  { v ,  N }  e.  P )
2726ralrimiva 2789 . . 3  |-  ( ( V  e.  W  /\  N  e.  V )  ->  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  P )
28 preq1 3883 . . . . 5  |-  ( x  =  v  ->  { x ,  N }  =  {
v ,  N }
)
2928eleq1d 2502 . . . 4  |-  ( x  =  v  ->  ( { x ,  N }  e.  P  <->  { v ,  N }  e.  P
) )
3029cbvralv 2932 . . 3  |-  ( A. x  e.  ( V  \  { N } ) { x ,  N }  e.  P  <->  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  P )
3127, 30sylibr 204 . 2  |-  ( ( V  e.  W  /\  N  e.  V )  ->  A. x  e.  ( V  \  { N } ) { x ,  N }  e.  P
)
32 simpl 444 . . . . . . . . . . 11  |-  ( ( a  =/=  N  /\  e  =  { a ,  N } )  -> 
a  =/=  N )
3332anim2i 553 . . . . . . . . . 10  |-  ( ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) )  ->  (
a  e.  V  /\  a  =/=  N ) )
3433adantl 453 . . . . . . . . 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 3927 . . . . . . . . 9  |-  ( a  e.  ( V  \  { N } )  <->  ( a  e.  V  /\  a  =/=  N ) )
3634, 35sylibr 204 . . . . . . . 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 2442 . . . . . . . . . . . . . 14  |-  ( e  =  { a ,  N }  ->  (
e  =  { x ,  N }  <->  { a ,  N }  =  {
x ,  N }
) )
3837adantl 453 . . . . . . . . . . . . 13  |-  ( ( a  =/=  N  /\  e  =  { a ,  N } )  -> 
( e  =  {
x ,  N }  <->  { a ,  N }  =  { x ,  N } ) )
3938ad2antlr 708 . . . . . . . . . . . 12  |-  ( ( ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) )  /\  x  e.  ( V  \  { N } ) )  -> 
( e  =  {
x ,  N }  <->  { a ,  N }  =  { x ,  N } ) )
40 vex 2959 . . . . . . . . . . . . . 14  |-  a  e. 
_V
41 vex 2959 . . . . . . . . . . . . . 14  |-  x  e. 
_V
4240, 41preqr1 3972 . . . . . . . . . . . . 13  |-  ( { a ,  N }  =  { x ,  N }  ->  a  =  x )
4342eqcomd 2441 . . . . . . . . . . . 12  |-  ( { a ,  N }  =  { x ,  N }  ->  x  =  a )
4439, 43syl6bi 220 . . . . . . . . . . 11  |-  ( ( ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) )  /\  x  e.  ( V  \  { N } ) )  -> 
( e  =  {
x ,  N }  ->  x  =  a ) )
4544adantll 695 . . . . . . . . . 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 3883 . . . . . . . . . . . . . . . 16  |-  ( a  =  x  ->  { a ,  N }  =  { x ,  N } )
4746equcoms 1693 . . . . . . . . . . . . . . 15  |-  ( x  =  a  ->  { a ,  N }  =  { x ,  N } )
4847eqeq2d 2447 . . . . . . . . . . . . . 14  |-  ( x  =  a  ->  (
e  =  { a ,  N }  <->  e  =  { x ,  N } ) )
4948biimpcd 216 . . . . . . . . . . . . 13  |-  ( e  =  { a ,  N }  ->  (
x  =  a  -> 
e  =  { x ,  N } ) )
5049adantl 453 . . . . . . . . . . . 12  |-  ( ( a  =/=  N  /\  e  =  { a ,  N } )  -> 
( x  =  a  ->  e  =  {
x ,  N }
) )
5150adantl 453 . . . . . . . . . . 11  |-  ( ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) )  ->  (
x  =  a  -> 
e  =  { x ,  N } ) )
5251ad2antlr 708 . . . . . . . . . 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 184 . . . . . . . . 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 2789 . . . . . . . 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 519 . . . . . . 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 424 . . . . . 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 2815 . . . . 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 587 . . . 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 2442 . . . . . . 7  |-  ( x  =  e  ->  (
x  =  { a ,  N }  <->  e  =  { a ,  N } ) )
6059anbi2d 685 . . . . . 6  |-  ( x  =  e  ->  (
( a  =/=  N  /\  x  =  {
a ,  N }
)  <->  ( a  =/= 
N  /\  e  =  { a ,  N } ) ) )
6160rexbidv 2726 . . . . 5  |-  ( x  =  e  ->  ( E. a  e.  V  ( a  =/=  N  /\  x  =  {
a ,  N }
)  <->  E. a  e.  V  ( a  =/=  N  /\  e  =  {
a ,  N }
) ) )
6261, 24elrab2 3094 . . . 4  |-  ( e  e.  P  <->  ( e  e.  ~P V  /\  E. a  e.  V  (
a  =/=  N  /\  e  =  { a ,  N } ) ) )
63 reu6 3123 . . . 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 262 . . 3  |-  ( ( V  e.  W  /\  N  e.  V )  ->  ( e  e.  P  ->  E! x  e.  ( V  \  { N } ) e  =  { x ,  N } ) )
6564ralrimiv 2788 . 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 5891 . 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 646 1  |-  ( ( V  e.  W  /\  N  e.  V )  ->  F : ( V 
\  { N }
)
-1-1-onto-> P )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   E!wreu 2707   {crab 2709    \ cdif 3317   ~Pcpw 3799   {csn 3814   {cpr 3815    e. cmpt 4266   -1-1-onto->wf1o 5453
This theorem is referenced by:  cusgrafilem3  21490
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462
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