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Theorem bj-inftyexpiinv 35011
Description: Utility theorem for the inverse of inftyexpi. (Contributed by BJ, 22-Jun-2019.) This utility theorem is irrelevant and should generally not be used. (New usage is discouraged.)
Assertion
Ref Expression
bj-inftyexpiinv  |-  ( A  e.  ( -u pi (,] pi )  ->  ( 1st `  (inftyexpi  `  A ) )  =  A )

Proof of Theorem bj-inftyexpiinv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 opeq1 4203 . . . 4  |-  ( x  =  A  ->  <. x ,  CC >.  =  <. A ,  CC >. )
2 df-bj-inftyexpi 35010 . . . 4  |- inftyexpi  =  ( x  e.  ( -u pi (,] pi )  |->  <.
x ,  CC >. )
3 opex 4701 . . . 4  |-  <. A ,  CC >.  e.  _V
41, 2, 3fvmpt 5931 . . 3  |-  ( A  e.  ( -u pi (,] pi )  ->  (inftyexpi  `  A )  =  <. A ,  CC >. )
54fveq2d 5852 . 2  |-  ( A  e.  ( -u pi (,] pi )  ->  ( 1st `  (inftyexpi  `  A ) )  =  ( 1st `  <. A ,  CC >. ) )
6 cnex 9562 . . 3  |-  CC  e.  _V
7 op1stg 6785 . . 3  |-  ( ( A  e.  ( -u pi (,] pi )  /\  CC  e.  _V )  -> 
( 1st `  <. A ,  CC >. )  =  A )
86, 7mpan2 669 . 2  |-  ( A  e.  ( -u pi (,] pi )  ->  ( 1st `  <. A ,  CC >. )  =  A )
95, 8eqtrd 2495 1  |-  ( A  e.  ( -u pi (,] pi )  ->  ( 1st `  (inftyexpi  `  A ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   _Vcvv 3106   <.cop 4022   ` cfv 5570  (class class class)co 6270   1stc1st 6771   CCcc 9479   -ucneg 9797   (,]cioc 11533   picpi 13884  inftyexpi cinftyexpi 35009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578  df-1st 6773  df-bj-inftyexpi 35010
This theorem is referenced by:  bj-inftyexpiinj  35012
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