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Theorem 1stpreimas 27705
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 6759 . . . . . . . . 9  |-  ( z  e.  ( _V  X.  _V )  <->  z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
21biimpi 194 . . . . . . . 8  |-  ( z  e.  ( _V  X.  _V )  ->  z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >. )
32ad2antrl 725 . . . . . . 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 5801 . . . . . . . . . . . 12  |-  ( 1st `  z )  e.  _V
54elsnc 3985 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  { X }  <->  ( 1st `  z )  =  X )
65biimpi 194 . . . . . . . . . 10  |-  ( ( 1st `  z )  e.  { X }  ->  ( 1st `  z
)  =  X )
76ad2antrl 725 . . . . . . . . 9  |-  ( ( z  e.  ( _V 
X.  _V )  /\  (
( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) )  -> 
( 1st `  z
)  =  X )
87adantl 464 . . . . . . . 8  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) ) )  ->  ( 1st `  z
)  =  X )
98opeq1d 4154 . . . . . . 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 2437 . . . . . 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 753 . . . . . . 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 764 . . . . . . 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 5292 . . . . . . . 8  |-  ( ( X  e.  V  /\  ( 2nd `  z )  e.  ( A " { X } ) )  ->  ( ( 2nd `  z )  e.  ( A " { X } )  <->  <. X , 
( 2nd `  z
) >.  e.  A ) )
1413biimpa 482 . . . . . . 7  |-  ( ( ( X  e.  V  /\  ( 2nd `  z
)  e.  ( A
" { X }
) )  /\  ( 2nd `  z )  e.  ( A " { X } ) )  ->  <. X ,  ( 2nd `  z ) >.  e.  A
)
1511, 12, 12, 14syl21anc 1225 . . . . . 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 2484 . . . . 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 5805 . . . . . . 7  |-  ( z  e.  A  ->  (
( 1st  |`  A ) `
 z )  =  ( 1st `  z
) )
1816, 17syl 16 . . . . . 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 2437 . . . . 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 530 . . . 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 4937 . . . . . . . . 9  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
2221biimpi 194 . . . . . . . 8  |-  ( Rel 
A  ->  A  C_  ( _V  X.  _V ) )
2322adantr 463 . . . . . . 7  |-  ( ( Rel  A  /\  X  e.  V )  ->  A  C_  ( _V  X.  _V ) )
2423sselda 3434 . . . . . 6  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  z  e.  A
)  ->  z  e.  ( _V  X.  _V )
)
2524adantrr 714 . . . . 5  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
z  e.  ( _V 
X.  _V ) )
2617ad2antrl 725 . . . . . . . 8  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( ( 1st  |`  A ) `
 z )  =  ( 1st `  z
) )
27 simprr 755 . . . . . . . 8  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( ( 1st  |`  A ) `
 z )  =  X )
2826, 27eqtr3d 2439 . . . . . . 7  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( 1st `  z
)  =  X )
2928, 5sylibr 212 . . . . . 6  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( 1st `  z
)  e.  { X } )
3028, 29eqeltrrd 2485 . . . . . . . . 9  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  ->  X  e.  { X } )
31 simpr 459 . . . . . . . . . . 11  |-  ( ( ( ( Rel  A  /\  X  e.  V
)  /\  ( z  e.  A  /\  (
( 1st  |`  A ) `
 z )  =  X ) )  /\  x  =  X )  ->  x  =  X )
3231opeq1d 4154 . . . . . . . . . 10  |-  ( ( ( ( Rel  A  /\  X  e.  V
)  /\  ( z  e.  A  /\  (
( 1st  |`  A ) `
 z )  =  X ) )  /\  x  =  X )  -> 
<. x ,  ( 2nd `  z ) >.  =  <. X ,  ( 2nd `  z
) >. )
3332eleq1d 2465 . . . . . . . . 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 6767 . . . . . . . . . . . 12  |-  ( ( Rel  A  /\  z  e.  A )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
3534ad2ant2r 744 . . . . . . . . . . 11  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
3628opeq1d 4154 . . . . . . . . . . 11  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  ->  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  =  <. X ,  ( 2nd `  z
) >. )
3735, 36eqtrd 2437 . . . . . . . . . 10  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
z  =  <. X , 
( 2nd `  z
) >. )
38 simprl 754 . . . . . . . . . 10  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
z  e.  A )
3937, 38eqeltrrd 2485 . . . . . . . . 9  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  ->  <. X ,  ( 2nd `  z ) >.  e.  A
)
4030, 33, 39rspcedvd 3157 . . . . . . . 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 2752 . . . . . . . 8  |-  ( E. x  e.  { X } <. x ,  ( 2nd `  z )
>.  e.  A  <->  E. x
( x  e.  { X }  /\  <. x ,  ( 2nd `  z
) >.  e.  A ) )
4240, 41sylib 196 . . . . . . 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 5801 . . . . . . . 8  |-  ( 2nd `  z )  e.  _V
4443elima3 5273 . . . . . . 7  |-  ( ( 2nd `  z )  e.  ( A " { X } )  <->  E. x
( x  e.  { X }  /\  <. x ,  ( 2nd `  z
) >.  e.  A ) )
4542, 44sylibr 212 . . . . . 6  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( 2nd `  z
)  e.  ( A
" { X }
) )
4629, 45jca 530 . . . . 5  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) )
4725, 46jca 530 . . . 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 830 . . 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 6754 . . . 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 6741 . . . . . . 7  |-  1st : _V -onto-> _V
52 fofn 5722 . . . . . . 7  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
5351, 52ax-mp 5 . . . . . 6  |-  1st  Fn  _V
54 ssv 3454 . . . . . 6  |-  A  C_  _V
55 fnssres 5619 . . . . . 6  |-  ( ( 1st  Fn  _V  /\  A  C_  _V )  -> 
( 1st  |`  A )  Fn  A )
5653, 54, 55mp2an 670 . . . . 5  |-  ( 1st  |`  A )  Fn  A
57 fniniseg 5927 . . . . 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 286 . 2  |-  ( ( Rel  A  /\  X  e.  V )  ->  (
z  e.  ( `' ( 1st  |`  A )
" { X }
)  <->  z  e.  ( { X }  X.  ( A " { X } ) ) ) )
6160eqrdv 2393 1  |-  ( ( Rel  A  /\  X  e.  V )  ->  ( `' ( 1st  |`  A )
" { X }
)  =  ( { X }  X.  ( A " { X }
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399   E.wex 1627    e. wcel 1836   E.wrex 2747   _Vcvv 3051    C_ wss 3406   {csn 3961   <.cop 3967    X. cxp 4928   `'ccnv 4929    |` cres 4932   "cima 4933   Rel wrel 4935    Fn wfn 5508   -onto->wfo 5511   ` cfv 5513   1stc1st 6719   2ndc2nd 6720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-fo 5519  df-fv 5521  df-1st 6721  df-2nd 6722
This theorem is referenced by:  gsummpt2d  27959
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