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Theorem 1stpreimas 28288
Description: The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.)
Assertion
Ref Expression
1stpreimas  |-  ( ( Rel  A  /\  X  e.  V )  ->  ( `' ( 1st  |`  A )
" { X }
)  =  ( { X }  X.  ( A " { X }
) ) )

Proof of Theorem 1stpreimas
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1st2ndb 6845 . . . . . . . . 9  |-  ( z  e.  ( _V  X.  _V )  <->  z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
21biimpi 197 . . . . . . . 8  |-  ( z  e.  ( _V  X.  _V )  ->  z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >. )
32ad2antrl 732 . . . . . . 7  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) ) )  ->  z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
4 fvex 5891 . . . . . . . . . . . 12  |-  ( 1st `  z )  e.  _V
54elsnc 4022 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  { X }  <->  ( 1st `  z )  =  X )
65biimpi 197 . . . . . . . . . 10  |-  ( ( 1st `  z )  e.  { X }  ->  ( 1st `  z
)  =  X )
76ad2antrl 732 . . . . . . . . 9  |-  ( ( z  e.  ( _V 
X.  _V )  /\  (
( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) )  -> 
( 1st `  z
)  =  X )
87adantl 467 . . . . . . . 8  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) ) )  ->  ( 1st `  z
)  =  X )
98opeq1d 4193 . . . . . . 7  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) ) )  ->  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  =  <. X ,  ( 2nd `  z
) >. )
103, 9eqtrd 2463 . . . . . 6  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) ) )  ->  z  =  <. X ,  ( 2nd `  z
) >. )
11 simplr 760 . . . . . . 7  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) ) )  ->  X  e.  V
)
12 simprrr 773 . . . . . . 7  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) ) )  ->  ( 2nd `  z
)  e.  ( A
" { X }
) )
13 elimasng 5213 . . . . . . . 8  |-  ( ( X  e.  V  /\  ( 2nd `  z )  e.  ( A " { X } ) )  ->  ( ( 2nd `  z )  e.  ( A " { X } )  <->  <. X , 
( 2nd `  z
) >.  e.  A ) )
1413biimpa 486 . . . . . . 7  |-  ( ( ( X  e.  V  /\  ( 2nd `  z
)  e.  ( A
" { X }
) )  /\  ( 2nd `  z )  e.  ( A " { X } ) )  ->  <. X ,  ( 2nd `  z ) >.  e.  A
)
1511, 12, 12, 14syl21anc 1263 . . . . . 6  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) ) )  ->  <. X ,  ( 2nd `  z )
>.  e.  A )
1610, 15eqeltrd 2507 . . . . 5  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) ) )  ->  z  e.  A
)
17 fvres 5895 . . . . . . 7  |-  ( z  e.  A  ->  (
( 1st  |`  A ) `
 z )  =  ( 1st `  z
) )
1816, 17syl 17 . . . . . 6  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) ) )  ->  ( ( 1st  |`  A ) `  z
)  =  ( 1st `  z ) )
1918, 8eqtrd 2463 . . . . 5  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) ) )  ->  ( ( 1st  |`  A ) `  z
)  =  X )
2016, 19jca 534 . . . 4  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) ) )  ->  ( z  e.  A  /\  ( ( 1st  |`  A ) `  z )  =  X ) )
21 df-rel 4860 . . . . . . . . 9  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
2221biimpi 197 . . . . . . . 8  |-  ( Rel 
A  ->  A  C_  ( _V  X.  _V ) )
2322adantr 466 . . . . . . 7  |-  ( ( Rel  A  /\  X  e.  V )  ->  A  C_  ( _V  X.  _V ) )
2423sselda 3464 . . . . . 6  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  z  e.  A
)  ->  z  e.  ( _V  X.  _V )
)
2524adantrr 721 . . . . 5  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
z  e.  ( _V 
X.  _V ) )
2617ad2antrl 732 . . . . . . . 8  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( ( 1st  |`  A ) `
 z )  =  ( 1st `  z
) )
27 simprr 764 . . . . . . . 8  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( ( 1st  |`  A ) `
 z )  =  X )
2826, 27eqtr3d 2465 . . . . . . 7  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( 1st `  z
)  =  X )
2928, 5sylibr 215 . . . . . 6  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( 1st `  z
)  e.  { X } )
3028, 29eqeltrrd 2508 . . . . . . . . 9  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  ->  X  e.  { X } )
31 simpr 462 . . . . . . . . . . 11  |-  ( ( ( ( Rel  A  /\  X  e.  V
)  /\  ( z  e.  A  /\  (
( 1st  |`  A ) `
 z )  =  X ) )  /\  x  =  X )  ->  x  =  X )
3231opeq1d 4193 . . . . . . . . . 10  |-  ( ( ( ( Rel  A  /\  X  e.  V
)  /\  ( z  e.  A  /\  (
( 1st  |`  A ) `
 z )  =  X ) )  /\  x  =  X )  -> 
<. x ,  ( 2nd `  z ) >.  =  <. X ,  ( 2nd `  z
) >. )
3332eleq1d 2491 . . . . . . . . 9  |-  ( ( ( ( Rel  A  /\  X  e.  V
)  /\  ( z  e.  A  /\  (
( 1st  |`  A ) `
 z )  =  X ) )  /\  x  =  X )  ->  ( <. x ,  ( 2nd `  z )
>.  e.  A  <->  <. X , 
( 2nd `  z
) >.  e.  A ) )
34 1st2nd 6853 . . . . . . . . . . . 12  |-  ( ( Rel  A  /\  z  e.  A )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
3534ad2ant2r 751 . . . . . . . . . . 11  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
3628opeq1d 4193 . . . . . . . . . . 11  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  ->  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  =  <. X ,  ( 2nd `  z
) >. )
3735, 36eqtrd 2463 . . . . . . . . . 10  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
z  =  <. X , 
( 2nd `  z
) >. )
38 simprl 762 . . . . . . . . . 10  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
z  e.  A )
3937, 38eqeltrrd 2508 . . . . . . . . 9  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  ->  <. X ,  ( 2nd `  z ) >.  e.  A
)
4030, 33, 39rspcedvd 3187 . . . . . . . 8  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  ->  E. x  e.  { X } <. x ,  ( 2nd `  z )
>.  e.  A )
41 df-rex 2777 . . . . . . . 8  |-  ( E. x  e.  { X } <. x ,  ( 2nd `  z )
>.  e.  A  <->  E. x
( x  e.  { X }  /\  <. x ,  ( 2nd `  z
) >.  e.  A ) )
4240, 41sylib 199 . . . . . . 7  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  ->  E. x ( x  e. 
{ X }  /\  <.
x ,  ( 2nd `  z ) >.  e.  A
) )
43 fvex 5891 . . . . . . . 8  |-  ( 2nd `  z )  e.  _V
4443elima3 5194 . . . . . . 7  |-  ( ( 2nd `  z )  e.  ( A " { X } )  <->  E. x
( x  e.  { X }  /\  <. x ,  ( 2nd `  z
) >.  e.  A ) )
4542, 44sylibr 215 . . . . . 6  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( 2nd `  z
)  e.  ( A
" { X }
) )
4629, 45jca 534 . . . . 5  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) )
4725, 46jca 534 . . . 4  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) ) )
4820, 47impbida 840 . . 3  |-  ( ( Rel  A  /\  X  e.  V )  ->  (
( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) )  <->  ( z  e.  A  /\  (
( 1st  |`  A ) `
 z )  =  X ) ) )
49 elxp7 6840 . . . 4  |-  ( z  e.  ( { X }  X.  ( A " { X } ) )  <-> 
( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) ) )
5049a1i 11 . . 3  |-  ( ( Rel  A  /\  X  e.  V )  ->  (
z  e.  ( { X }  X.  ( A " { X }
) )  <->  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z )  e.  { X }  /\  ( 2nd `  z )  e.  ( A " { X } ) ) ) ) )
51 fo1st 6827 . . . . . . 7  |-  1st : _V -onto-> _V
52 fofn 5812 . . . . . . 7  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
5351, 52ax-mp 5 . . . . . 6  |-  1st  Fn  _V
54 ssv 3484 . . . . . 6  |-  A  C_  _V
55 fnssres 5707 . . . . . 6  |-  ( ( 1st  Fn  _V  /\  A  C_  _V )  -> 
( 1st  |`  A )  Fn  A )
5653, 54, 55mp2an 676 . . . . 5  |-  ( 1st  |`  A )  Fn  A
57 fniniseg 6018 . . . . 5  |-  ( ( 1st  |`  A )  Fn  A  ->  ( z  e.  ( `' ( 1st  |`  A ) " { X } )  <-> 
( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) ) )
5856, 57ax-mp 5 . . . 4  |-  ( z  e.  ( `' ( 1st  |`  A ) " { X } )  <-> 
( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )
5958a1i 11 . . 3  |-  ( ( Rel  A  /\  X  e.  V )  ->  (
z  e.  ( `' ( 1st  |`  A )
" { X }
)  <->  ( z  e.  A  /\  ( ( 1st  |`  A ) `  z )  =  X ) ) )
6048, 50, 593bitr4rd 289 . 2  |-  ( ( Rel  A  /\  X  e.  V )  ->  (
z  e.  ( `' ( 1st  |`  A )
" { X }
)  <->  z  e.  ( { X }  X.  ( A " { X } ) ) ) )
6160eqrdv 2419 1  |-  ( ( Rel  A  /\  X  e.  V )  ->  ( `' ( 1st  |`  A )
" { X }
)  =  ( { X }  X.  ( A " { X }
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   E.wrex 2772   _Vcvv 3080    C_ wss 3436   {csn 3998   <.cop 4004    X. cxp 4851   `'ccnv 4852    |` cres 4855   "cima 4856   Rel wrel 4858    Fn wfn 5596   -onto->wfo 5599   ` cfv 5601   1stc1st 6805   2ndc2nd 6806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fo 5607  df-fv 5609  df-1st 6807  df-2nd 6808
This theorem is referenced by:  gsummpt2d  28551
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