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Theorem djuf1olem 6906
Description: Lemma for djulf1o 6911 and djurf1o 6912. (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 3526 . . . . 5  |-  X  e. 
{ X }
4 opelxpi 4541 . . . . 5  |-  ( ( X  e.  { X }  /\  x  e.  A
)  ->  <. X ,  x >.  e.  ( { X }  X.  A
) )
53, 4mpan 420 . . . 4  |-  ( x  e.  A  ->  <. X ,  x >.  e.  ( { X }  X.  A
) )
65adantl 275 . . 3  |-  ( ( T.  /\  x  e.  A )  ->  <. X ,  x >.  e.  ( { X }  X.  A
) )
7 xp2nd 6032 . . . 4  |-  ( y  e.  ( { X }  X.  A )  -> 
( 2nd `  y
)  e.  A )
87adantl 275 . . 3  |-  ( ( T.  /\  y  e.  ( { X }  X.  A ) )  -> 
( 2nd `  y
)  e.  A )
9 1st2nd2 6041 . . . . . . . 8  |-  ( y  e.  ( { X }  X.  A )  -> 
y  =  <. ( 1st `  y ) ,  ( 2nd `  y
) >. )
10 xp1st 6031 . . . . . . . . . 10  |-  ( y  e.  ( { X }  X.  A )  -> 
( 1st `  y
)  e.  { X } )
11 elsni 3515 . . . . . . . . . 10  |-  ( ( 1st `  y )  e.  { X }  ->  ( 1st `  y
)  =  X )
1210, 11syl 14 . . . . . . . . 9  |-  ( y  e.  ( { X }  X.  A )  -> 
( 1st `  y
)  =  X )
1312opeq1d 3681 . . . . . . . 8  |-  ( y  e.  ( { X }  X.  A )  ->  <. ( 1st `  y
) ,  ( 2nd `  y ) >.  =  <. X ,  ( 2nd `  y
) >. )
149, 13eqtrd 2150 . . . . . . 7  |-  ( y  e.  ( { X }  X.  A )  -> 
y  =  <. X , 
( 2nd `  y
) >. )
1514eqeq2d 2129 . . . . . 6  |-  ( y  e.  ( { X }  X.  A )  -> 
( <. X ,  x >.  =  y  <->  <. X ,  x >.  =  <. X , 
( 2nd `  y
) >. ) )
16 eqcom 2119 . . . . . 6  |-  ( <. X ,  x >.  =  y  <->  y  =  <. X ,  x >. )
17 eqid 2117 . . . . . . 7  |-  X  =  X
18 vex 2663 . . . . . . . 8  |-  x  e. 
_V
192, 18opth 4129 . . . . . . 7  |-  ( <. X ,  x >.  = 
<. X ,  ( 2nd `  y ) >.  <->  ( X  =  X  /\  x  =  ( 2nd `  y
) ) )
2017, 19mpbiran 909 . . . . . 6  |-  ( <. X ,  x >.  = 
<. X ,  ( 2nd `  y ) >.  <->  x  =  ( 2nd `  y ) )
2115, 16, 203bitr3g 221 . . . . 5  |-  ( y  e.  ( { X }  X.  A )  -> 
( y  =  <. X ,  x >.  <->  x  =  ( 2nd `  y ) ) )
2221bicomd 140 . . . 4  |-  ( y  e.  ( { X }  X.  A )  -> 
( x  =  ( 2nd `  y )  <-> 
y  =  <. X ,  x >. ) )
2322ad2antll 482 . . 3  |-  ( ( T.  /\  ( x  e.  A  /\  y  e.  ( { X }  X.  A ) ) )  ->  ( x  =  ( 2nd `  y
)  <->  y  =  <. X ,  x >. )
)
241, 6, 8, 23f1o2d 5943 . 2  |-  ( T. 
->  F : A -1-1-onto-> ( { X }  X.  A
) )
2524mptru 1325 1  |-  F : A
-1-1-onto-> ( { X }  X.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1316   T. wtru 1317    e. wcel 1465   _Vcvv 2660   {csn 3497   <.cop 3500    |-> cmpt 3959    X. cxp 4507   -1-1-onto->wf1o 5092   ` cfv 5093   1stc1st 6004   2ndc2nd 6005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007
This theorem is referenced by:  djuf1olemr  6907  djulf1o  6911  djurf1o  6912
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