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Theorem 1stpreimas 28338
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 6863 . . . . . . . . 9  |-  ( z  e.  ( _V  X.  _V )  <->  z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
21biimpi 199 . . . . . . . 8  |-  ( z  e.  ( _V  X.  _V )  ->  z  = 
<. ( 1st `  z
) ,  ( 2nd `  z ) >. )
32ad2antrl 739 . . . . . . 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 5902 . . . . . . . . . . . 12  |-  ( 1st `  z )  e.  _V
54elsnc 4004 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  { X }  <->  ( 1st `  z )  =  X )
65biimpi 199 . . . . . . . . . 10  |-  ( ( 1st `  z )  e.  { X }  ->  ( 1st `  z
)  =  X )
76ad2antrl 739 . . . . . . . . 9  |-  ( ( z  e.  ( _V 
X.  _V )  /\  (
( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) )  -> 
( 1st `  z
)  =  X )
87adantl 472 . . . . . . . 8  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) ) )  ->  ( 1st `  z
)  =  X )
98opeq1d 4186 . . . . . . 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 2496 . . . . . 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 767 . . . . . . 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 780 . . . . . . 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 5216 . . . . . . . 8  |-  ( ( X  e.  V  /\  ( 2nd `  z )  e.  ( A " { X } ) )  ->  ( ( 2nd `  z )  e.  ( A " { X } )  <->  <. X , 
( 2nd `  z
) >.  e.  A ) )
1413biimpa 491 . . . . . . 7  |-  ( ( ( X  e.  V  /\  ( 2nd `  z
)  e.  ( A
" { X }
) )  /\  ( 2nd `  z )  e.  ( A " { X } ) )  ->  <. X ,  ( 2nd `  z ) >.  e.  A
)
1511, 12, 12, 14syl21anc 1275 . . . . . 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 2540 . . . . 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 5906 . . . . . . 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 2496 . . . . 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 539 . . . 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 4863 . . . . . . . . 9  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
2221biimpi 199 . . . . . . . 8  |-  ( Rel 
A  ->  A  C_  ( _V  X.  _V ) )
2322adantr 471 . . . . . . 7  |-  ( ( Rel  A  /\  X  e.  V )  ->  A  C_  ( _V  X.  _V ) )
2423sselda 3444 . . . . . 6  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  z  e.  A
)  ->  z  e.  ( _V  X.  _V )
)
2524adantrr 728 . . . . 5  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
z  e.  ( _V 
X.  _V ) )
2617ad2antrl 739 . . . . . . . 8  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( ( 1st  |`  A ) `
 z )  =  ( 1st `  z
) )
27 simprr 771 . . . . . . . 8  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( ( 1st  |`  A ) `
 z )  =  X )
2826, 27eqtr3d 2498 . . . . . . 7  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( 1st `  z
)  =  X )
2928, 5sylibr 217 . . . . . 6  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( 1st `  z
)  e.  { X } )
3028, 29eqeltrrd 2541 . . . . . . . . 9  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  ->  X  e.  { X } )
31 simpr 467 . . . . . . . . . . 11  |-  ( ( ( ( Rel  A  /\  X  e.  V
)  /\  ( z  e.  A  /\  (
( 1st  |`  A ) `
 z )  =  X ) )  /\  x  =  X )  ->  x  =  X )
3231opeq1d 4186 . . . . . . . . . 10  |-  ( ( ( ( Rel  A  /\  X  e.  V
)  /\  ( z  e.  A  /\  (
( 1st  |`  A ) `
 z )  =  X ) )  /\  x  =  X )  -> 
<. x ,  ( 2nd `  z ) >.  =  <. X ,  ( 2nd `  z
) >. )
3332eleq1d 2524 . . . . . . . . 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 6871 . . . . . . . . . . . 12  |-  ( ( Rel  A  /\  z  e.  A )  ->  z  =  <. ( 1st `  z
) ,  ( 2nd `  z ) >. )
3534ad2ant2r 758 . . . . . . . . . . 11  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
z  =  <. ( 1st `  z ) ,  ( 2nd `  z
) >. )
3628opeq1d 4186 . . . . . . . . . . 11  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  ->  <. ( 1st `  z
) ,  ( 2nd `  z ) >.  =  <. X ,  ( 2nd `  z
) >. )
3735, 36eqtrd 2496 . . . . . . . . . 10  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
z  =  <. X , 
( 2nd `  z
) >. )
38 simprl 769 . . . . . . . . . 10  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
z  e.  A )
3937, 38eqeltrrd 2541 . . . . . . . . 9  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  ->  <. X ,  ( 2nd `  z ) >.  e.  A
)
4030, 33, 39rspcedvd 3167 . . . . . . . 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 2755 . . . . . . . 8  |-  ( E. x  e.  { X } <. x ,  ( 2nd `  z )
>.  e.  A  <->  E. x
( x  e.  { X }  /\  <. x ,  ( 2nd `  z
) >.  e.  A ) )
4240, 41sylib 201 . . . . . . 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 5902 . . . . . . . 8  |-  ( 2nd `  z )  e.  _V
4443elima3 5197 . . . . . . 7  |-  ( ( 2nd `  z )  e.  ( A " { X } )  <->  E. x
( x  e.  { X }  /\  <. x ,  ( 2nd `  z
) >.  e.  A ) )
4542, 44sylibr 217 . . . . . 6  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( 2nd `  z
)  e.  ( A
" { X }
) )
4629, 45jca 539 . . . . 5  |-  ( ( ( Rel  A  /\  X  e.  V )  /\  ( z  e.  A  /\  ( ( 1st  |`  A ) `
 z )  =  X ) )  -> 
( ( 1st `  z
)  e.  { X }  /\  ( 2nd `  z
)  e.  ( A
" { X }
) ) )
4725, 46jca 539 . . . 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 848 . . 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 6858 . . . 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 6845 . . . . . . 7  |-  1st : _V -onto-> _V
52 fofn 5822 . . . . . . 7  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
5351, 52ax-mp 5 . . . . . 6  |-  1st  Fn  _V
54 ssv 3464 . . . . . 6  |-  A  C_  _V
55 fnssres 5715 . . . . . 6  |-  ( ( 1st  Fn  _V  /\  A  C_  _V )  -> 
( 1st  |`  A )  Fn  A )
5653, 54, 55mp2an 683 . . . . 5  |-  ( 1st  |`  A )  Fn  A
57 fniniseg 6031 . . . . 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 294 . 2  |-  ( ( Rel  A  /\  X  e.  V )  ->  (
z  e.  ( `' ( 1st  |`  A )
" { X }
)  <->  z  e.  ( { X }  X.  ( A " { X } ) ) ) )
6160eqrdv 2460 1  |-  ( ( Rel  A  /\  X  e.  V )  ->  ( `' ( 1st  |`  A )
" { X }
)  =  ( { X }  X.  ( A " { X }
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1455   E.wex 1674    e. wcel 1898   E.wrex 2750   _Vcvv 3057    C_ wss 3416   {csn 3980   <.cop 3986    X. cxp 4854   `'ccnv 4855    |` cres 4858   "cima 4859   Rel wrel 4861    Fn wfn 5600   -onto->wfo 5603   ` cfv 5605   1stc1st 6823   2ndc2nd 6824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pow 4598  ax-pr 4656  ax-un 6615
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4419  df-opab 4478  df-mpt 4479  df-id 4771  df-xp 4862  df-rel 4863  df-cnv 4864  df-co 4865  df-dm 4866  df-rn 4867  df-res 4868  df-ima 4869  df-iota 5569  df-fun 5607  df-fn 5608  df-f 5609  df-fo 5611  df-fv 5613  df-1st 6825  df-2nd 6826
This theorem is referenced by:  gsummpt2d  28595
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