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Theorem djuf1olem 6745
Description: Lemma for djulf1o 6750 and djurf1o 6751. (Contributed by BJ and Jim Kingdon, 4-Jul-2022.)
Hypotheses
Ref Expression
djuf1olem.1  |-  X  e. 
_V
djuf1olem.2  |-  F  =  ( x  e.  A  |-> 
<. X ,  x >. )
Assertion
Ref Expression
djuf1olem  |-  F : A
-1-1-onto-> ( { X }  X.  A )
Distinct variable groups:    x, X    x, A
Allowed substitution hint:    F( x)

Proof of Theorem djuf1olem
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 djuf1olem.2 . . 3  |-  F  =  ( x  e.  A  |-> 
<. X ,  x >. )
2 djuf1olem.1 . . . . . 6  |-  X  e. 
_V
32snid 3475 . . . . 5  |-  X  e. 
{ X }
4 opelxpi 4469 . . . . 5  |-  ( ( X  e.  { X }  /\  x  e.  A
)  ->  <. X ,  x >.  e.  ( { X }  X.  A
) )
53, 4mpan 415 . . . 4  |-  ( x  e.  A  ->  <. X ,  x >.  e.  ( { X }  X.  A
) )
65adantl 271 . . 3  |-  ( ( T.  /\  x  e.  A )  ->  <. X ,  x >.  e.  ( { X }  X.  A
) )
7 xp2nd 5937 . . . 4  |-  ( y  e.  ( { X }  X.  A )  -> 
( 2nd `  y
)  e.  A )
87adantl 271 . . 3  |-  ( ( T.  /\  y  e.  ( { X }  X.  A ) )  -> 
( 2nd `  y
)  e.  A )
9 1st2nd2 5945 . . . . . . . 8  |-  ( y  e.  ( { X }  X.  A )  -> 
y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
10 xp1st 5936 . . . . . . . . . 10  |-  ( y  e.  ( { X }  X.  A )  -> 
( 1st `  y
)  e.  { X } )
11 elsni 3464 . . . . . . . . . 10  |-  ( ( 1st `  y )  e.  { X }  ->  ( 1st `  y
)  =  X )
1210, 11syl 14 . . . . . . . . 9  |-  ( y  e.  ( { X }  X.  A )  -> 
( 1st `  y
)  =  X )
1312opeq1d 3628 . . . . . . . 8  |-  ( y  e.  ( { X }  X.  A )  ->  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  =  <. X ,  ( 2nd `  y
) >. )
149, 13eqtrd 2120 . . . . . . 7  |-  ( y  e.  ( { X }  X.  A )  -> 
y  =  <. X , 
( 2nd `  y
) >. )
1514eqeq2d 2099 . . . . . 6  |-  ( y  e.  ( { X }  X.  A )  -> 
( <. X ,  x >.  =  y  <->  <. X ,  x >.  =  <. X , 
( 2nd `  y
) >. ) )
16 eqcom 2090 . . . . . 6  |-  ( <. X ,  x >.  =  y  <->  y  =  <. X ,  x >. )
17 eqid 2088 . . . . . . 7  |-  X  =  X
18 vex 2622 . . . . . . . 8  |-  x  e. 
_V
192, 18opth 4064 . . . . . . 7  |-  ( <. X ,  x >.  = 
<. X ,  ( 2nd `  y ) >.  <->  ( X  =  X  /\  x  =  ( 2nd `  y
) ) )
2017, 19mpbiran 886 . . . . . 6  |-  ( <. X ,  x >.  = 
<. X ,  ( 2nd `  y ) >.  <->  x  =  ( 2nd `  y ) )
2115, 16, 203bitr3g 220 . . . . 5  |-  ( y  e.  ( { X }  X.  A )  -> 
( y  =  <. X ,  x >.  <->  x  =  ( 2nd `  y ) ) )
2221bicomd 139 . . . 4  |-  ( y  e.  ( { X }  X.  A )  -> 
( x  =  ( 2nd `  y )  <-> 
y  =  <. X ,  x >. ) )
2322ad2antll 475 . . 3  |-  ( ( T.  /\  ( x  e.  A  /\  y  e.  ( { X }  X.  A ) ) )  ->  ( x  =  ( 2nd `  y
)  <->  y  =  <. X ,  x >. )
)
241, 6, 8, 23f1o2d 5849 . 2  |-  ( T. 
->  F : A -1-1-onto-> ( { X }  X.  A
) )
2524mptru 1298 1  |-  F : A
-1-1-onto-> ( { X }  X.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1289   T. wtru 1290    e. wcel 1438   _Vcvv 2619   {csn 3446   <.cop 3449    |-> cmpt 3899    X. cxp 4436   -1-1-onto->wf1o 5014   ` cfv 5015   1stc1st 5909   2ndc2nd 5910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-1st 5911  df-2nd 5912
This theorem is referenced by:  djuf1olemr  6746  djulf1o  6750  djurf1o  6751
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