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Theorem ixpsnf1o 6533
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 snexg 4040 . . . 4  |-  ( I  e.  V  ->  { I }  e.  _V )
3 vex 2636 . . . . 5  |-  x  e. 
_V
43snex 4041 . . . 4  |-  { x }  e.  _V
5 xpexg 4581 . . . 4  |-  ( ( { I }  e.  _V  /\  { x }  e.  _V )  ->  ( { I }  X.  { x } )  e.  _V )
62, 4, 5sylancl 405 . . 3  |-  ( I  e.  V  ->  ( { I }  X.  { x } )  e.  _V )
76adantr 271 . 2  |-  ( ( I  e.  V  /\  x  e.  A )  ->  ( { I }  X.  { x } )  e.  _V )
8 vex 2636 . . . . 5  |-  a  e. 
_V
98rnex 4732 . . . 4  |-  ran  a  e.  _V
109uniex 4288 . . 3  |-  U. ran  a  e.  _V
1110a1i 9 . 2  |-  ( ( I  e.  V  /\  a  e.  X_ y  e. 
{ I } A
)  ->  U. ran  a  e.  _V )
12 sneq 3477 . . . . . 6  |-  ( b  =  I  ->  { b }  =  { I } )
1312xpeq1d 4490 . . . . 5  |-  ( b  =  I  ->  ( { b }  X.  { x } )  =  ( { I }  X.  { x }
) )
1413eqeq2d 2106 . . . 4  |-  ( b  =  I  ->  (
a  =  ( { b }  X.  {
x } )  <->  a  =  ( { I }  X.  { x } ) ) )
1514anbi2d 453 . . 3  |-  ( b  =  I  ->  (
( x  e.  A  /\  a  =  ( { b }  X.  { x } ) )  <->  ( x  e.  A  /\  a  =  ( { I }  X.  { x } ) ) ) )
16 elixpsn 6532 . . . . . 6  |-  ( b  e.  _V  ->  (
a  e.  X_ y  e.  { b } A  <->  E. c  e.  A  a  =  { <. b ,  c >. } ) )
1716elv 2637 . . . . 5  |-  ( a  e.  X_ y  e.  {
b } A  <->  E. c  e.  A  a  =  { <. b ,  c
>. } )
1812ixpeq1d 6507 . . . . . 6  |-  ( b  =  I  ->  X_ y  e.  { b } A  =  X_ y  e.  {
I } A )
1918eleq2d 2164 . . . . 5  |-  ( b  =  I  ->  (
a  e.  X_ y  e.  { b } A  <->  a  e.  X_ y  e.  {
I } A ) )
2017, 19syl5bbr 193 . . . 4  |-  ( b  =  I  ->  ( E. c  e.  A  a  =  { <. b ,  c >. }  <->  a  e.  X_ y  e.  { I } A ) )
2120anbi1d 454 . . 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
) ) )
22 vex 2636 . . . . . . 7  |-  b  e. 
_V
2322, 3xpsn 5512 . . . . . 6  |-  ( { b }  X.  {
x } )  =  { <. b ,  x >. }
2423eqeq2i 2105 . . . . 5  |-  ( a  =  ( { b }  X.  { x } )  <->  a  =  { <. b ,  x >. } )
2524anbi2i 446 . . . 4  |-  ( ( x  e.  A  /\  a  =  ( {
b }  X.  {
x } ) )  <-> 
( x  e.  A  /\  a  =  { <. b ,  x >. } ) )
26 eqid 2095 . . . . . . . . 9  |-  { <. b ,  x >. }  =  { <. b ,  x >. }
27 opeq2 3645 . . . . . . . . . . 11  |-  ( c  =  x  ->  <. b ,  c >.  =  <. b ,  x >. )
2827sneqd 3479 . . . . . . . . . 10  |-  ( c  =  x  ->  { <. b ,  c >. }  =  { <. b ,  x >. } )
2928rspceeqv 2753 . . . . . . . . 9  |-  ( ( x  e.  A  /\  {
<. b ,  x >. }  =  { <. b ,  x >. } )  ->  E. c  e.  A  { <. b ,  x >. }  =  { <. b ,  c >. } )
3026, 29mpan2 417 . . . . . . . 8  |-  ( x  e.  A  ->  E. c  e.  A  { <. b ,  x >. }  =  { <. b ,  c >. } )
3122, 3op2nda 4949 . . . . . . . . 9  |-  U. ran  {
<. b ,  x >. }  =  x
3231eqcomi 2099 . . . . . . . 8  |-  x  = 
U. ran  { <. b ,  x >. }
3330, 32jctir 307 . . . . . . 7  |-  ( x  e.  A  ->  ( E. c  e.  A  { <. b ,  x >. }  =  { <. b ,  c >. }  /\  x  =  U. ran  { <. b ,  x >. } ) )
34 eqeq1 2101 . . . . . . . . 9  |-  ( a  =  { <. b ,  x >. }  ->  (
a  =  { <. b ,  c >. }  <->  { <. b ,  x >. }  =  { <. b ,  c >. } ) )
3534rexbidv 2392 . . . . . . . 8  |-  ( a  =  { <. b ,  x >. }  ->  ( E. c  e.  A  a  =  { <. b ,  c >. }  <->  E. c  e.  A  { <. b ,  x >. }  =  { <. b ,  c >. } ) )
36 rneq 4694 . . . . . . . . . 10  |-  ( a  =  { <. b ,  x >. }  ->  ran  a  =  ran  { <. b ,  x >. } )
3736unieqd 3686 . . . . . . . . 9  |-  ( a  =  { <. b ,  x >. }  ->  U. ran  a  =  U. ran  { <. b ,  x >. } )
3837eqeq2d 2106 . . . . . . . 8  |-  ( a  =  { <. b ,  x >. }  ->  (
x  =  U. ran  a 
<->  x  =  U. ran  {
<. b ,  x >. } ) )
3935, 38anbi12d 458 . . . . . . 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 156 . . . . . 6  |-  ( x  e.  A  ->  (
a  =  { <. b ,  x >. }  ->  ( E. c  e.  A  a  =  { <. b ,  c >. }  /\  x  =  U. ran  a
) ) )
4140imp 123 . . . . 5  |-  ( ( x  e.  A  /\  a  =  { <. b ,  x >. } )  -> 
( E. c  e.  A  a  =  { <. b ,  c >. }  /\  x  =  U. ran  a ) )
42 vex 2636 . . . . . . . . . . 11  |-  c  e. 
_V
4322, 42op2nda 4949 . . . . . . . . . 10  |-  U. ran  {
<. b ,  c >. }  =  c
4443eqeq2i 2105 . . . . . . . . 9  |-  ( x  =  U. ran  { <. b ,  c >. } 
<->  x  =  c )
45 eqidd 2096 . . . . . . . . . . 11  |-  ( c  e.  A  ->  { <. b ,  c >. }  =  { <. b ,  c
>. } )
4645ancli 317 . . . . . . . . . 10  |-  ( c  e.  A  ->  (
c  e.  A  /\  {
<. b ,  c >. }  =  { <. b ,  c >. } ) )
47 eleq1w 2155 . . . . . . . . . . 11  |-  ( x  =  c  ->  (
x  e.  A  <->  c  e.  A ) )
48 opeq2 3645 . . . . . . . . . . . . 13  |-  ( x  =  c  ->  <. b ,  x >.  =  <. b ,  c >. )
4948sneqd 3479 . . . . . . . . . . . 12  |-  ( x  =  c  ->  { <. b ,  x >. }  =  { <. b ,  c
>. } )
5049eqeq2d 2106 . . . . . . . . . . 11  |-  ( x  =  c  ->  ( { <. b ,  c
>. }  =  { <. b ,  x >. }  <->  { <. b ,  c >. }  =  { <. b ,  c
>. } ) )
5147, 50anbi12d 458 . . . . . . . . . 10  |-  ( x  =  c  ->  (
( x  e.  A  /\  { <. b ,  c
>. }  =  { <. b ,  x >. } )  <-> 
( c  e.  A  /\  { <. b ,  c
>. }  =  { <. b ,  c >. } ) ) )
5246, 51syl5ibrcom 156 . . . . . . . . 9  |-  ( c  e.  A  ->  (
x  =  c  -> 
( x  e.  A  /\  { <. b ,  c
>. }  =  { <. b ,  x >. } ) ) )
5344, 52syl5bi 151 . . . . . . . 8  |-  ( c  e.  A  ->  (
x  =  U. ran  {
<. b ,  c >. }  ->  ( x  e.  A  /\  { <. b ,  c >. }  =  { <. b ,  x >. } ) ) )
54 rneq 4694 . . . . . . . . . . 11  |-  ( a  =  { <. b ,  c >. }  ->  ran  a  =  ran  { <. b ,  c >. } )
5554unieqd 3686 . . . . . . . . . 10  |-  ( a  =  { <. b ,  c >. }  ->  U.
ran  a  =  U. ran  { <. b ,  c
>. } )
5655eqeq2d 2106 . . . . . . . . 9  |-  ( a  =  { <. b ,  c >. }  ->  ( x  =  U. ran  a 
<->  x  =  U. ran  {
<. b ,  c >. } ) )
57 eqeq1 2101 . . . . . . . . . 10  |-  ( a  =  { <. b ,  c >. }  ->  ( a  =  { <. b ,  x >. }  <->  { <. b ,  c >. }  =  { <. b ,  x >. } ) )
5857anbi2d 453 . . . . . . . . 9  |-  ( a  =  { <. b ,  c >. }  ->  ( ( x  e.  A  /\  a  =  { <. b ,  x >. } )  <->  ( x  e.  A  /\  { <. b ,  c >. }  =  { <. b ,  x >. } ) ) )
5956, 58imbi12d 233 . . . . . . . 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 156 . . . . . . 7  |-  ( c  e.  A  ->  (
a  =  { <. b ,  c >. }  ->  ( x  =  U. ran  a  ->  ( x  e.  A  /\  a  =  { <. b ,  x >. } ) ) ) )
6160rexlimiv 2496 . . . . . 6  |-  ( E. c  e.  A  a  =  { <. b ,  c >. }  ->  ( x  =  U. ran  a  ->  ( x  e.  A  /\  a  =  { <. b ,  x >. } ) ) )
6261imp 123 . . . . 5  |-  ( ( E. c  e.  A  a  =  { <. b ,  c >. }  /\  x  =  U. ran  a
)  ->  ( x  e.  A  /\  a  =  { <. b ,  x >. } ) )
6341, 62impbii 125 . . . 4  |-  ( ( x  e.  A  /\  a  =  { <. b ,  x >. } )  <->  ( E. c  e.  A  a  =  { <. b ,  c
>. }  /\  x  = 
U. ran  a )
)
6425, 63bitri 183 . . 3  |-  ( ( x  e.  A  /\  a  =  ( {
b }  X.  {
x } ) )  <-> 
( E. c  e.  A  a  =  { <. b ,  c >. }  /\  x  =  U. ran  a ) )
6515, 21, 64vtoclbg 2694 . 2  |-  ( I  e.  V  ->  (
( x  e.  A  /\  a  =  ( { I }  X.  { x } ) )  <->  ( a  e.  X_ y  e.  { I } A  /\  x  =  U. ran  a ) ) )
661, 7, 11, 65f1od 5885 1  |-  ( I  e.  V  ->  F : A -1-1-onto-> X_ y  e.  {
I } A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1296    e. wcel 1445   E.wrex 2371   _Vcvv 2633   {csn 3466   <.cop 3469   U.cuni 3675    |-> cmpt 3921    X. cxp 4465   ran crn 4468   -1-1-onto->wf1o 5048   X_cixp 6495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-reu 2377  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ixp 6496
This theorem is referenced by:  mapsnf1o  6534
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