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Theorem ixpsnf1o 7580
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 4641 . . . 4  |-  { I }  e.  _V
3 snex 4641 . . . 4  |-  { x }  e.  _V
42, 3xpex 6614 . . 3  |-  ( { I }  X.  {
x } )  e. 
_V
54a1i 11 . 2  |-  ( ( I  e.  V  /\  x  e.  A )  ->  ( { I }  X.  { x } )  e.  _V )
6 vex 3034 . . . . 5  |-  a  e. 
_V
76rnex 6746 . . . 4  |-  ran  a  e.  _V
87uniex 6606 . . 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 3969 . . . . . 6  |-  ( b  =  I  ->  { b }  =  { I } )
1110xpeq1d 4862 . . . . 5  |-  ( b  =  I  ->  ( { b }  X.  { x } )  =  ( { I }  X.  { x }
) )
1211eqeq2d 2481 . . . 4  |-  ( b  =  I  ->  (
a  =  ( { b }  X.  {
x } )  <->  a  =  ( { I }  X.  { x } ) ) )
1312anbi2d 718 . . 3  |-  ( b  =  I  ->  (
( x  e.  A  /\  a  =  ( { b }  X.  { x } ) )  <->  ( x  e.  A  /\  a  =  ( { I }  X.  { x } ) ) ) )
14 vex 3034 . . . . . 6  |-  b  e. 
_V
15 elixpsn 7579 . . . . . 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 7552 . . . . . 6  |-  ( b  =  I  ->  X_ y  e.  { b } A  =  X_ y  e.  {
I } A )
1817eleq2d 2534 . . . . 5  |-  ( b  =  I  ->  (
a  e.  X_ y  e.  { b } A  <->  a  e.  X_ y  e.  {
I } A ) )
1916, 18syl5bbr 267 . . . 4  |-  ( b  =  I  ->  ( E. c  e.  A  a  =  { <. b ,  c >. }  <->  a  e.  X_ y  e.  { I } A ) )
2019anbi1d 719 . . 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 3034 . . . . . . 7  |-  x  e. 
_V
2214, 21xpsn 6082 . . . . . 6  |-  ( { b }  X.  {
x } )  =  { <. b ,  x >. }
2322eqeq2i 2483 . . . . 5  |-  ( a  =  ( { b }  X.  { x } )  <->  a  =  { <. b ,  x >. } )
2423anbi2i 708 . . . 4  |-  ( ( x  e.  A  /\  a  =  ( {
b }  X.  {
x } ) )  <-> 
( x  e.  A  /\  a  =  { <. b ,  x >. } ) )
25 eqid 2471 . . . . . . . . 9  |-  { <. b ,  x >. }  =  { <. b ,  x >. }
26 opeq2 4159 . . . . . . . . . . . 12  |-  ( c  =  x  ->  <. b ,  c >.  =  <. b ,  x >. )
2726sneqd 3971 . . . . . . . . . . 11  |-  ( c  =  x  ->  { <. b ,  c >. }  =  { <. b ,  x >. } )
2827eqeq2d 2481 . . . . . . . . . 10  |-  ( c  =  x  ->  ( { <. b ,  x >. }  =  { <. b ,  c >. }  <->  { <. b ,  x >. }  =  { <. b ,  x >. } ) )
2928rspcev 3136 . . . . . . . . 9  |-  ( ( x  e.  A  /\  {
<. b ,  x >. }  =  { <. b ,  x >. } )  ->  E. c  e.  A  { <. b ,  x >. }  =  { <. b ,  c >. } )
3025, 29mpan2 685 . . . . . . . 8  |-  ( x  e.  A  ->  E. c  e.  A  { <. b ,  x >. }  =  { <. b ,  c >. } )
3114, 21op2nda 5328 . . . . . . . . 9  |-  U. ran  {
<. b ,  x >. }  =  x
3231eqcomi 2480 . . . . . . . 8  |-  x  = 
U. ran  { <. b ,  x >. }
3330, 32jctir 547 . . . . . . 7  |-  ( x  e.  A  ->  ( E. c  e.  A  { <. b ,  x >. }  =  { <. b ,  c >. }  /\  x  =  U. ran  { <. b ,  x >. } ) )
34 eqeq1 2475 . . . . . . . . 9  |-  ( a  =  { <. b ,  x >. }  ->  (
a  =  { <. b ,  c >. }  <->  { <. b ,  x >. }  =  { <. b ,  c >. } ) )
3534rexbidv 2892 . . . . . . . 8  |-  ( a  =  { <. b ,  x >. }  ->  ( E. c  e.  A  a  =  { <. b ,  c >. }  <->  E. c  e.  A  { <. b ,  x >. }  =  { <. b ,  c >. } ) )
36 rneq 5066 . . . . . . . . . 10  |-  ( a  =  { <. b ,  x >. }  ->  ran  a  =  ran  { <. b ,  x >. } )
3736unieqd 4200 . . . . . . . . 9  |-  ( a  =  { <. b ,  x >. }  ->  U. ran  a  =  U. ran  { <. b ,  x >. } )
3837eqeq2d 2481 . . . . . . . 8  |-  ( a  =  { <. b ,  x >. }  ->  (
x  =  U. ran  a 
<->  x  =  U. ran  {
<. b ,  x >. } ) )
3935, 38anbi12d 725 . . . . . . 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 230 . . . . . 6  |-  ( x  e.  A  ->  (
a  =  { <. b ,  x >. }  ->  ( E. c  e.  A  a  =  { <. b ,  c >. }  /\  x  =  U. ran  a
) ) )
4140imp 436 . . . . 5  |-  ( ( x  e.  A  /\  a  =  { <. b ,  x >. } )  -> 
( E. c  e.  A  a  =  { <. b ,  c >. }  /\  x  =  U. ran  a ) )
42 vex 3034 . . . . . . . . . . 11  |-  c  e. 
_V
4314, 42op2nda 5328 . . . . . . . . . 10  |-  U. ran  {
<. b ,  c >. }  =  c
4443eqeq2i 2483 . . . . . . . . 9  |-  ( x  =  U. ran  { <. b ,  c >. } 
<->  x  =  c )
45 eqidd 2472 . . . . . . . . . . 11  |-  ( c  e.  A  ->  { <. b ,  c >. }  =  { <. b ,  c
>. } )
4645ancli 560 . . . . . . . . . 10  |-  ( c  e.  A  ->  (
c  e.  A  /\  {
<. b ,  c >. }  =  { <. b ,  c >. } ) )
47 eleq1 2537 . . . . . . . . . . 11  |-  ( x  =  c  ->  (
x  e.  A  <->  c  e.  A ) )
48 opeq2 4159 . . . . . . . . . . . . 13  |-  ( x  =  c  ->  <. b ,  x >.  =  <. b ,  c >. )
4948sneqd 3971 . . . . . . . . . . . 12  |-  ( x  =  c  ->  { <. b ,  x >. }  =  { <. b ,  c
>. } )
5049eqeq2d 2481 . . . . . . . . . . 11  |-  ( x  =  c  ->  ( { <. b ,  c
>. }  =  { <. b ,  x >. }  <->  { <. b ,  c >. }  =  { <. b ,  c
>. } ) )
5147, 50anbi12d 725 . . . . . . . . . 10  |-  ( x  =  c  ->  (
( x  e.  A  /\  { <. b ,  c
>. }  =  { <. b ,  x >. } )  <-> 
( c  e.  A  /\  { <. b ,  c
>. }  =  { <. b ,  c >. } ) ) )
5246, 51syl5ibrcom 230 . . . . . . . . 9  |-  ( c  e.  A  ->  (
x  =  c  -> 
( x  e.  A  /\  { <. b ,  c
>. }  =  { <. b ,  x >. } ) ) )
5344, 52syl5bi 225 . . . . . . . 8  |-  ( c  e.  A  ->  (
x  =  U. ran  {
<. b ,  c >. }  ->  ( x  e.  A  /\  { <. b ,  c >. }  =  { <. b ,  x >. } ) ) )
54 rneq 5066 . . . . . . . . . . 11  |-  ( a  =  { <. b ,  c >. }  ->  ran  a  =  ran  { <. b ,  c >. } )
5554unieqd 4200 . . . . . . . . . 10  |-  ( a  =  { <. b ,  c >. }  ->  U.
ran  a  =  U. ran  { <. b ,  c
>. } )
5655eqeq2d 2481 . . . . . . . . 9  |-  ( a  =  { <. b ,  c >. }  ->  ( x  =  U. ran  a 
<->  x  =  U. ran  {
<. b ,  c >. } ) )
57 eqeq1 2475 . . . . . . . . . 10  |-  ( a  =  { <. b ,  c >. }  ->  ( a  =  { <. b ,  x >. }  <->  { <. b ,  c >. }  =  { <. b ,  x >. } ) )
5857anbi2d 718 . . . . . . . . 9  |-  ( a  =  { <. b ,  c >. }  ->  ( ( x  e.  A  /\  a  =  { <. b ,  x >. } )  <->  ( x  e.  A  /\  { <. b ,  c >. }  =  { <. b ,  x >. } ) ) )
5956, 58imbi12d 327 . . . . . . . 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 230 . . . . . . 7  |-  ( c  e.  A  ->  (
a  =  { <. b ,  c >. }  ->  ( x  =  U. ran  a  ->  ( x  e.  A  /\  a  =  { <. b ,  x >. } ) ) ) )
6160rexlimiv 2867 . . . . . 6  |-  ( E. c  e.  A  a  =  { <. b ,  c >. }  ->  ( x  =  U. ran  a  ->  ( x  e.  A  /\  a  =  { <. b ,  x >. } ) ) )
6261imp 436 . . . . 5  |-  ( ( E. c  e.  A  a  =  { <. b ,  c >. }  /\  x  =  U. ran  a
)  ->  ( x  e.  A  /\  a  =  { <. b ,  x >. } ) )
6341, 62impbii 192 . . . 4  |-  ( ( x  e.  A  /\  a  =  { <. b ,  x >. } )  <->  ( E. c  e.  A  a  =  { <. b ,  c
>. }  /\  x  = 
U. ran  a )
)
6424, 63bitri 257 . . 3  |-  ( ( x  e.  A  /\  a  =  ( {
b }  X.  {
x } ) )  <-> 
( E. c  e.  A  a  =  { <. b ,  c >. }  /\  x  =  U. ran  a ) )
6513, 20, 64vtoclbg 3094 . 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 6538 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 189    /\ wa 376    = wceq 1452    e. wcel 1904   E.wrex 2757   _Vcvv 3031   {csn 3959   <.cop 3965   U.cuni 4190    |-> cmpt 4454    X. cxp 4837   ran crn 4840   -1-1-onto->wf1o 5588   X_cixp 7540
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-8 1906  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-pow 4579  ax-pr 4639  ax-un 6602
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  df-ixp 7541
This theorem is referenced by:  mapsnf1o  7581
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