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Theorem ixpsnf1o 7303
Description: A bijection between a class and single-point functions to it. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypothesis
Ref Expression
ixpsnf1o.f  |-  F  =  ( x  e.  A  |->  ( { I }  X.  { x } ) )
Assertion
Ref Expression
ixpsnf1o  |-  ( I  e.  V  ->  F : A -1-1-onto-> X_ y  e.  {
I } A )
Distinct variable groups:    x, I,
y    x, A, y    x, V, y    y, F
Allowed substitution hint:    F( x)

Proof of Theorem ixpsnf1o
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ixpsnf1o.f . 2  |-  F  =  ( x  e.  A  |->  ( { I }  X.  { x } ) )
2 snex 4533 . . . 4  |-  { I }  e.  _V
3 snex 4533 . . . 4  |-  { x }  e.  _V
42, 3xpex 6508 . . 3  |-  ( { I }  X.  {
x } )  e. 
_V
54a1i 11 . 2  |-  ( ( I  e.  V  /\  x  e.  A )  ->  ( { I }  X.  { x } )  e.  _V )
6 vex 2975 . . . . 5  |-  a  e. 
_V
76rnex 6512 . . . 4  |-  ran  a  e.  _V
87uniex 6376 . . 3  |-  U. ran  a  e.  _V
98a1i 11 . 2  |-  ( ( I  e.  V  /\  a  e.  X_ y  e. 
{ I } A
)  ->  U. ran  a  e.  _V )
10 sneq 3887 . . . . . 6  |-  ( b  =  I  ->  { b }  =  { I } )
1110xpeq1d 4863 . . . . 5  |-  ( b  =  I  ->  ( { b }  X.  { x } )  =  ( { I }  X.  { x }
) )
1211eqeq2d 2454 . . . 4  |-  ( b  =  I  ->  (
a  =  ( { b }  X.  {
x } )  <->  a  =  ( { I }  X.  { x } ) ) )
1312anbi2d 703 . . 3  |-  ( b  =  I  ->  (
( x  e.  A  /\  a  =  ( { b }  X.  { x } ) )  <->  ( x  e.  A  /\  a  =  ( { I }  X.  { x } ) ) ) )
14 vex 2975 . . . . . 6  |-  b  e. 
_V
15 elixpsn 7302 . . . . . 6  |-  ( b  e.  _V  ->  (
a  e.  X_ y  e.  { b } A  <->  E. c  e.  A  a  =  { <. b ,  c >. } ) )
1614, 15ax-mp 5 . . . . 5  |-  ( a  e.  X_ y  e.  {
b } A  <->  E. c  e.  A  a  =  { <. b ,  c
>. } )
1710ixpeq1d 7275 . . . . . 6  |-  ( b  =  I  ->  X_ y  e.  { b } A  =  X_ y  e.  {
I } A )
1817eleq2d 2510 . . . . 5  |-  ( b  =  I  ->  (
a  e.  X_ y  e.  { b } A  <->  a  e.  X_ y  e.  {
I } A ) )
1916, 18syl5bbr 259 . . . 4  |-  ( b  =  I  ->  ( E. c  e.  A  a  =  { <. b ,  c >. }  <->  a  e.  X_ y  e.  { I } A ) )
2019anbi1d 704 . . 3  |-  ( b  =  I  ->  (
( E. c  e.  A  a  =  { <. b ,  c >. }  /\  x  =  U. ran  a )  <->  ( a  e.  X_ y  e.  {
I } A  /\  x  =  U. ran  a
) ) )
21 vex 2975 . . . . . . 7  |-  x  e. 
_V
2214, 21xpsn 5885 . . . . . 6  |-  ( { b }  X.  {
x } )  =  { <. b ,  x >. }
2322eqeq2i 2453 . . . . 5  |-  ( a  =  ( { b }  X.  { x } )  <->  a  =  { <. b ,  x >. } )
2423anbi2i 694 . . . 4  |-  ( ( x  e.  A  /\  a  =  ( {
b }  X.  {
x } ) )  <-> 
( x  e.  A  /\  a  =  { <. b ,  x >. } ) )
25 eqid 2443 . . . . . . . . 9  |-  { <. b ,  x >. }  =  { <. b ,  x >. }
26 opeq2 4060 . . . . . . . . . . . 12  |-  ( c  =  x  ->  <. b ,  c >.  =  <. b ,  x >. )
2726sneqd 3889 . . . . . . . . . . 11  |-  ( c  =  x  ->  { <. b ,  c >. }  =  { <. b ,  x >. } )
2827eqeq2d 2454 . . . . . . . . . 10  |-  ( c  =  x  ->  ( { <. b ,  x >. }  =  { <. b ,  c >. }  <->  { <. b ,  x >. }  =  { <. b ,  x >. } ) )
2928rspcev 3073 . . . . . . . . 9  |-  ( ( x  e.  A  /\  {
<. b ,  x >. }  =  { <. b ,  x >. } )  ->  E. c  e.  A  { <. b ,  x >. }  =  { <. b ,  c >. } )
3025, 29mpan2 671 . . . . . . . 8  |-  ( x  e.  A  ->  E. c  e.  A  { <. b ,  x >. }  =  { <. b ,  c >. } )
3114, 21op2nda 5324 . . . . . . . . 9  |-  U. ran  {
<. b ,  x >. }  =  x
3231eqcomi 2447 . . . . . . . 8  |-  x  = 
U. ran  { <. b ,  x >. }
3330, 32jctir 538 . . . . . . 7  |-  ( x  e.  A  ->  ( E. c  e.  A  { <. b ,  x >. }  =  { <. b ,  c >. }  /\  x  =  U. ran  { <. b ,  x >. } ) )
34 eqeq1 2449 . . . . . . . . 9  |-  ( a  =  { <. b ,  x >. }  ->  (
a  =  { <. b ,  c >. }  <->  { <. b ,  x >. }  =  { <. b ,  c >. } ) )
3534rexbidv 2736 . . . . . . . 8  |-  ( a  =  { <. b ,  x >. }  ->  ( E. c  e.  A  a  =  { <. b ,  c >. }  <->  E. c  e.  A  { <. b ,  x >. }  =  { <. b ,  c >. } ) )
36 rneq 5065 . . . . . . . . . 10  |-  ( a  =  { <. b ,  x >. }  ->  ran  a  =  ran  { <. b ,  x >. } )
3736unieqd 4101 . . . . . . . . 9  |-  ( a  =  { <. b ,  x >. }  ->  U. ran  a  =  U. ran  { <. b ,  x >. } )
3837eqeq2d 2454 . . . . . . . 8  |-  ( a  =  { <. b ,  x >. }  ->  (
x  =  U. ran  a 
<->  x  =  U. ran  {
<. b ,  x >. } ) )
3935, 38anbi12d 710 . . . . . . 7  |-  ( a  =  { <. b ,  x >. }  ->  (
( E. c  e.  A  a  =  { <. b ,  c >. }  /\  x  =  U. ran  a )  <->  ( E. c  e.  A  { <. b ,  x >. }  =  { <. b ,  c >. }  /\  x  =  U. ran  { <. b ,  x >. } ) ) )
4033, 39syl5ibrcom 222 . . . . . 6  |-  ( x  e.  A  ->  (
a  =  { <. b ,  x >. }  ->  ( E. c  e.  A  a  =  { <. b ,  c >. }  /\  x  =  U. ran  a
) ) )
4140imp 429 . . . . 5  |-  ( ( x  e.  A  /\  a  =  { <. b ,  x >. } )  -> 
( E. c  e.  A  a  =  { <. b ,  c >. }  /\  x  =  U. ran  a ) )
42 vex 2975 . . . . . . . . . . 11  |-  c  e. 
_V
4314, 42op2nda 5324 . . . . . . . . . 10  |-  U. ran  {
<. b ,  c >. }  =  c
4443eqeq2i 2453 . . . . . . . . 9  |-  ( x  =  U. ran  { <. b ,  c >. } 
<->  x  =  c )
45 eqidd 2444 . . . . . . . . . . 11  |-  ( c  e.  A  ->  { <. b ,  c >. }  =  { <. b ,  c
>. } )
4645ancli 551 . . . . . . . . . 10  |-  ( c  e.  A  ->  (
c  e.  A  /\  {
<. b ,  c >. }  =  { <. b ,  c >. } ) )
47 eleq1 2503 . . . . . . . . . . 11  |-  ( x  =  c  ->  (
x  e.  A  <->  c  e.  A ) )
48 opeq2 4060 . . . . . . . . . . . . 13  |-  ( x  =  c  ->  <. b ,  x >.  =  <. b ,  c >. )
4948sneqd 3889 . . . . . . . . . . . 12  |-  ( x  =  c  ->  { <. b ,  x >. }  =  { <. b ,  c
>. } )
5049eqeq2d 2454 . . . . . . . . . . 11  |-  ( x  =  c  ->  ( { <. b ,  c
>. }  =  { <. b ,  x >. }  <->  { <. b ,  c >. }  =  { <. b ,  c
>. } ) )
5147, 50anbi12d 710 . . . . . . . . . 10  |-  ( x  =  c  ->  (
( x  e.  A  /\  { <. b ,  c
>. }  =  { <. b ,  x >. } )  <-> 
( c  e.  A  /\  { <. b ,  c
>. }  =  { <. b ,  c >. } ) ) )
5246, 51syl5ibrcom 222 . . . . . . . . 9  |-  ( c  e.  A  ->  (
x  =  c  -> 
( x  e.  A  /\  { <. b ,  c
>. }  =  { <. b ,  x >. } ) ) )
5344, 52syl5bi 217 . . . . . . . 8  |-  ( c  e.  A  ->  (
x  =  U. ran  {
<. b ,  c >. }  ->  ( x  e.  A  /\  { <. b ,  c >. }  =  { <. b ,  x >. } ) ) )
54 rneq 5065 . . . . . . . . . . 11  |-  ( a  =  { <. b ,  c >. }  ->  ran  a  =  ran  { <. b ,  c >. } )
5554unieqd 4101 . . . . . . . . . 10  |-  ( a  =  { <. b ,  c >. }  ->  U.
ran  a  =  U. ran  { <. b ,  c
>. } )
5655eqeq2d 2454 . . . . . . . . 9  |-  ( a  =  { <. b ,  c >. }  ->  ( x  =  U. ran  a 
<->  x  =  U. ran  {
<. b ,  c >. } ) )
57 eqeq1 2449 . . . . . . . . . 10  |-  ( a  =  { <. b ,  c >. }  ->  ( a  =  { <. b ,  x >. }  <->  { <. b ,  c >. }  =  { <. b ,  x >. } ) )
5857anbi2d 703 . . . . . . . . 9  |-  ( a  =  { <. b ,  c >. }  ->  ( ( x  e.  A  /\  a  =  { <. b ,  x >. } )  <->  ( x  e.  A  /\  { <. b ,  c >. }  =  { <. b ,  x >. } ) ) )
5956, 58imbi12d 320 . . . . . . . 8  |-  ( a  =  { <. b ,  c >. }  ->  ( ( x  =  U. ran  a  ->  ( x  e.  A  /\  a  =  { <. b ,  x >. } ) )  <->  ( x  =  U. ran  { <. b ,  c >. }  ->  ( x  e.  A  /\  {
<. b ,  c >. }  =  { <. b ,  x >. } ) ) ) )
6053, 59syl5ibrcom 222 . . . . . . 7  |-  ( c  e.  A  ->  (
a  =  { <. b ,  c >. }  ->  ( x  =  U. ran  a  ->  ( x  e.  A  /\  a  =  { <. b ,  x >. } ) ) ) )
6160rexlimiv 2835 . . . . . 6  |-  ( E. c  e.  A  a  =  { <. b ,  c >. }  ->  ( x  =  U. ran  a  ->  ( x  e.  A  /\  a  =  { <. b ,  x >. } ) ) )
6261imp 429 . . . . 5  |-  ( ( E. c  e.  A  a  =  { <. b ,  c >. }  /\  x  =  U. ran  a
)  ->  ( x  e.  A  /\  a  =  { <. b ,  x >. } ) )
6341, 62impbii 188 . . . 4  |-  ( ( x  e.  A  /\  a  =  { <. b ,  x >. } )  <->  ( E. c  e.  A  a  =  { <. b ,  c
>. }  /\  x  = 
U. ran  a )
)
6424, 63bitri 249 . . 3  |-  ( ( x  e.  A  /\  a  =  ( {
b }  X.  {
x } ) )  <-> 
( E. c  e.  A  a  =  { <. b ,  c >. }  /\  x  =  U. ran  a ) )
6513, 20, 64vtoclbg 3031 . 2  |-  ( I  e.  V  ->  (
( x  e.  A  /\  a  =  ( { I }  X.  { x } ) )  <->  ( a  e.  X_ y  e.  { I } A  /\  x  =  U. ran  a ) ) )
661, 5, 9, 65f1od 6310 1  |-  ( I  e.  V  ->  F : A -1-1-onto-> X_ y  e.  {
I } A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2716   _Vcvv 2972   {csn 3877   <.cop 3883   U.cuni 4091    e. cmpt 4350    X. cxp 4838   ran crn 4841   -1-1-onto->wf1o 5417   X_cixp 7263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ixp 7264
This theorem is referenced by:  mapsnf1o  7304
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