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Theorem cusgrfilem2 39682
Description: Lemma 2 for cusgrfi 39684. (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 11-Nov-2020.)
Hypotheses
Ref Expression
cusgrfi.v  |-  V  =  (Vtx `  G )
cusgrfi.p  |-  P  =  { x  e.  ~P V  |  E. a  e.  V  ( a  =/=  N  /\  x  =  { a ,  N } ) }
cusgrfi.f  |-  F  =  ( x  e.  ( V  \  { N } )  |->  { x ,  N } )
Assertion
Ref Expression
cusgrfilem2  |-  ( N  e.  V  ->  F : ( V  \  { N } ) -1-1-onto-> P )
Distinct variable groups:    x, G    N, a, x    V, a, x    x, P
Allowed substitution hints:    P( a)    F( x, a)    G( a)

Proof of Theorem cusgrfilem2
Dummy variables  e 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifi 3544 . . . . 5  |-  ( x  e.  ( V  \  { N } )  ->  x  e.  V )
2 id 22 . . . . 5  |-  ( N  e.  V  ->  N  e.  V )
3 prelpwi 4647 . . . . 5  |-  ( ( x  e.  V  /\  N  e.  V )  ->  { x ,  N }  e.  ~P V
)
41, 2, 3syl2anr 486 . . . 4  |-  ( ( N  e.  V  /\  x  e.  ( V  \  { N } ) )  ->  { x ,  N }  e.  ~P V )
51adantl 473 . . . . 5  |-  ( ( N  e.  V  /\  x  e.  ( V  \  { N } ) )  ->  x  e.  V )
6 eldifsni 4089 . . . . . . 7  |-  ( x  e.  ( V  \  { N } )  ->  x  =/=  N )
76adantl 473 . . . . . 6  |-  ( ( N  e.  V  /\  x  e.  ( V  \  { N } ) )  ->  x  =/=  N )
8 eqidd 2472 . . . . . 6  |-  ( ( N  e.  V  /\  x  e.  ( V  \  { N } ) )  ->  { x ,  N }  =  {
x ,  N }
)
97, 8jca 541 . . . . 5  |-  ( ( N  e.  V  /\  x  e.  ( V  \  { N } ) )  ->  ( x  =/=  N  /\  { x ,  N }  =  {
x ,  N }
) )
10 id 22 . . . . . 6  |-  ( x  e.  V  ->  x  e.  V )
11 neeq1 2705 . . . . . . . 8  |-  ( a  =  x  ->  (
a  =/=  N  <->  x  =/=  N ) )
12 preq1 4042 . . . . . . . . 9  |-  ( a  =  x  ->  { a ,  N }  =  { x ,  N } )
1312eqeq2d 2481 . . . . . . . 8  |-  ( a  =  x  ->  ( { x ,  N }  =  { a ,  N }  <->  { x ,  N }  =  {
x ,  N }
) )
1411, 13anbi12d 725 . . . . . . 7  |-  ( a  =  x  ->  (
( a  =/=  N  /\  { x ,  N }  =  { a ,  N } )  <->  ( x  =/=  N  /\  { x ,  N }  =  {
x ,  N }
) ) )
1514adantl 473 . . . . . 6  |-  ( ( x  e.  V  /\  a  =  x )  ->  ( ( a  =/= 
N  /\  { x ,  N }  =  {
a ,  N }
)  <->  ( x  =/= 
N  /\  { x ,  N }  =  {
x ,  N }
) ) )
1610, 15rspcedv 3140 . . . . 5  |-  ( x  e.  V  ->  (
( x  =/=  N  /\  { x ,  N }  =  { x ,  N } )  ->  E. a  e.  V  ( a  =/=  N  /\  { x ,  N }  =  { a ,  N } ) ) )
175, 9, 16sylc 61 . . . 4  |-  ( ( N  e.  V  /\  x  e.  ( V  \  { N } ) )  ->  E. a  e.  V  ( a  =/=  N  /\  { x ,  N }  =  {
a ,  N }
) )
18 cusgrfi.p . . . . . 6  |-  P  =  { x  e.  ~P V  |  E. a  e.  V  ( a  =/=  N  /\  x  =  { a ,  N } ) }
1918eleq2i 2541 . . . . 5  |-  ( { x ,  N }  e.  P  <->  { x ,  N }  e.  { x  e.  ~P V  |  E. a  e.  V  (
a  =/=  N  /\  x  =  { a ,  N } ) } )
20 eqeq1 2475 . . . . . . . 8  |-  ( v  =  { x ,  N }  ->  (
v  =  { a ,  N }  <->  { x ,  N }  =  {
a ,  N }
) )
2120anbi2d 718 . . . . . . 7  |-  ( v  =  { x ,  N }  ->  (
( a  =/=  N  /\  v  =  {
a ,  N }
)  <->  ( a  =/= 
N  /\  { x ,  N }  =  {
a ,  N }
) ) )
2221rexbidv 2892 . . . . . 6  |-  ( v  =  { x ,  N }  ->  ( E. a  e.  V  ( a  =/=  N  /\  v  =  {
a ,  N }
)  <->  E. a  e.  V  ( a  =/=  N  /\  { x ,  N }  =  { a ,  N } ) ) )
23 eqeq1 2475 . . . . . . . . 9  |-  ( x  =  v  ->  (
x  =  { a ,  N }  <->  v  =  { a ,  N } ) )
2423anbi2d 718 . . . . . . . 8  |-  ( x  =  v  ->  (
( a  =/=  N  /\  x  =  {
a ,  N }
)  <->  ( a  =/= 
N  /\  v  =  { a ,  N } ) ) )
2524rexbidv 2892 . . . . . . 7  |-  ( x  =  v  ->  ( E. a  e.  V  ( a  =/=  N  /\  x  =  {
a ,  N }
)  <->  E. a  e.  V  ( a  =/=  N  /\  v  =  {
a ,  N }
) ) )
2625cbvrabv 3030 . . . . . 6  |-  { x  e.  ~P V  |  E. a  e.  V  (
a  =/=  N  /\  x  =  { a ,  N } ) }  =  { v  e. 
~P V  |  E. a  e.  V  (
a  =/=  N  /\  v  =  { a ,  N } ) }
2722, 26elrab2 3186 . . . . 5  |-  ( { x ,  N }  e.  { x  e.  ~P V  |  E. a  e.  V  ( a  =/=  N  /\  x  =  { a ,  N } ) }  <->  ( {
x ,  N }  e.  ~P V  /\  E. a  e.  V  (
a  =/=  N  /\  { x ,  N }  =  { a ,  N } ) ) )
2819, 27bitri 257 . . . 4  |-  ( { x ,  N }  e.  P  <->  ( { x ,  N }  e.  ~P V  /\  E. a  e.  V  ( a  =/= 
N  /\  { x ,  N }  =  {
a ,  N }
) ) )
294, 17, 28sylanbrc 677 . . 3  |-  ( ( N  e.  V  /\  x  e.  ( V  \  { N } ) )  ->  { x ,  N }  e.  P
)
3029ralrimiva 2809 . 2  |-  ( N  e.  V  ->  A. x  e.  ( V  \  { N } ) { x ,  N }  e.  P
)
31 simpl 464 . . . . . . . . . . 11  |-  ( ( a  =/=  N  /\  e  =  { a ,  N } )  -> 
a  =/=  N )
3231anim2i 579 . . . . . . . . . 10  |-  ( ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) )  ->  (
a  e.  V  /\  a  =/=  N ) )
3332adantl 473 . . . . . . . . 9  |-  ( ( ( N  e.  V  /\  e  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) ) )  -> 
( a  e.  V  /\  a  =/=  N
) )
34 eldifsn 4088 . . . . . . . . 9  |-  ( a  e.  ( V  \  { N } )  <->  ( a  e.  V  /\  a  =/=  N ) )
3533, 34sylibr 217 . . . . . . . 8  |-  ( ( ( N  e.  V  /\  e  e.  ~P V )  /\  (
a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) ) )  -> 
a  e.  ( V 
\  { N }
) )
36 eqeq1 2475 . . . . . . . . . . . . . 14  |-  ( e  =  { a ,  N }  ->  (
e  =  { x ,  N }  <->  { a ,  N }  =  {
x ,  N }
) )
3736adantl 473 . . . . . . . . . . . . 13  |-  ( ( a  =/=  N  /\  e  =  { a ,  N } )  -> 
( e  =  {
x ,  N }  <->  { a ,  N }  =  { x ,  N } ) )
3837ad2antlr 741 . . . . . . . . . . . 12  |-  ( ( ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) )  /\  x  e.  ( V  \  { N } ) )  -> 
( e  =  {
x ,  N }  <->  { a ,  N }  =  { x ,  N } ) )
39 vex 3034 . . . . . . . . . . . . . 14  |-  a  e. 
_V
40 vex 3034 . . . . . . . . . . . . . 14  |-  x  e. 
_V
4139, 40preqr1 4139 . . . . . . . . . . . . 13  |-  ( { a ,  N }  =  { x ,  N }  ->  a  =  x )
4241eqcomd 2477 . . . . . . . . . . . 12  |-  ( { a ,  N }  =  { x ,  N }  ->  x  =  a )
4338, 42syl6bi 236 . . . . . . . . . . 11  |-  ( ( ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) )  /\  x  e.  ( V  \  { N } ) )  -> 
( e  =  {
x ,  N }  ->  x  =  a ) )
4443adantll 728 . . . . . . . . . 10  |-  ( ( ( ( N  e.  V  /\  e  e. 
~P V )  /\  ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) ) )  /\  x  e.  ( V  \  { N } ) )  ->  ( e  =  { x ,  N }  ->  x  =  a ) )
4512equcoms 1872 . . . . . . . . . . . . . . 15  |-  ( x  =  a  ->  { a ,  N }  =  { x ,  N } )
4645eqeq2d 2481 . . . . . . . . . . . . . 14  |-  ( x  =  a  ->  (
e  =  { a ,  N }  <->  e  =  { x ,  N } ) )
4746biimpcd 232 . . . . . . . . . . . . 13  |-  ( e  =  { a ,  N }  ->  (
x  =  a  -> 
e  =  { x ,  N } ) )
4847adantl 473 . . . . . . . . . . . 12  |-  ( ( a  =/=  N  /\  e  =  { a ,  N } )  -> 
( x  =  a  ->  e  =  {
x ,  N }
) )
4948adantl 473 . . . . . . . . . . 11  |-  ( ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) )  ->  (
x  =  a  -> 
e  =  { x ,  N } ) )
5049ad2antlr 741 . . . . . . . . . 10  |-  ( ( ( ( N  e.  V  /\  e  e. 
~P V )  /\  ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) ) )  /\  x  e.  ( V  \  { N } ) )  ->  ( x  =  a  ->  e  =  { x ,  N } ) )
5144, 50impbid 195 . . . . . . . . 9  |-  ( ( ( ( N  e.  V  /\  e  e. 
~P V )  /\  ( a  e.  V  /\  ( a  =/=  N  /\  e  =  {
a ,  N }
) ) )  /\  x  e.  ( V  \  { N } ) )  ->  ( e  =  { x ,  N } 
<->  x  =  a ) )
5251ralrimiva 2809 . . . . . . . 8  |-  ( ( ( 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 ) )
5335, 52jca 541 . . . . . . 7  |-  ( ( ( 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 ) ) )
5453ex 441 . . . . . 6  |-  ( ( 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 ) ) ) )
5554reximdv2 2855 . . . . 5  |-  ( ( 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 ) ) )
5655expimpd 614 . . . 4  |-  ( 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 ) ) )
57 eqeq1 2475 . . . . . . 7  |-  ( x  =  e  ->  (
x  =  { a ,  N }  <->  e  =  { a ,  N } ) )
5857anbi2d 718 . . . . . 6  |-  ( x  =  e  ->  (
( a  =/=  N  /\  x  =  {
a ,  N }
)  <->  ( a  =/= 
N  /\  e  =  { a ,  N } ) ) )
5958rexbidv 2892 . . . . 5  |-  ( x  =  e  ->  ( E. a  e.  V  ( a  =/=  N  /\  x  =  {
a ,  N }
)  <->  E. a  e.  V  ( a  =/=  N  /\  e  =  {
a ,  N }
) ) )
6059, 18elrab2 3186 . . . 4  |-  ( e  e.  P  <->  ( e  e.  ~P V  /\  E. a  e.  V  (
a  =/=  N  /\  e  =  { a ,  N } ) ) )
61 reu6 3215 . . . 4  |-  ( E! x  e.  ( V 
\  { N }
) e  =  {
x ,  N }  <->  E. a  e.  ( V 
\  { N }
) A. x  e.  ( V  \  { N } ) ( e  =  { x ,  N }  <->  x  =  a ) )
6256, 60, 613imtr4g 278 . . 3  |-  ( N  e.  V  ->  (
e  e.  P  ->  E! x  e.  ( V  \  { N }
) e  =  {
x ,  N }
) )
6362ralrimiv 2808 . 2  |-  ( N  e.  V  ->  A. e  e.  P  E! x  e.  ( V  \  { N } ) e  =  { x ,  N } )
64 cusgrfi.f . . 3  |-  F  =  ( x  e.  ( V  \  { N } )  |->  { x ,  N } )
6564f1ompt 6059 . 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 } ) )
6630, 63, 65sylanbrc 677 1  |-  ( N  e.  V  ->  F : ( V  \  { N } ) -1-1-onto-> P )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   E!wreu 2758   {crab 2760    \ cdif 3387   ~Pcpw 3942   {csn 3959   {cpr 3961    |-> cmpt 4454   -1-1-onto->wf1o 5588   ` cfv 5589  Vtxcvtx 39251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597
This theorem is referenced by:  cusgrfilem3  39683
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