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Theorem bj-inftyexpiinv 32840
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 4160 . . . 4  |-  ( x  =  A  ->  <. x ,  CC >.  =  <. A ,  CC >. )
2 df-bj-inftyexpi 32839 . . . 4  |- inftyexpi  =  ( x  e.  ( -u pi (,] pi )  |->  <.
x ,  CC >. )
3 opex 4657 . . . 4  |-  <. A ,  CC >.  e.  _V
41, 2, 3fvmpt 5876 . . 3  |-  ( A  e.  ( -u pi (,] pi )  ->  (inftyexpi  `  A )  =  <. A ,  CC >. )
54fveq2d 5796 . 2  |-  ( A  e.  ( -u pi (,] pi )  ->  ( 1st `  (inftyexpi  `  A ) )  =  ( 1st `  <. A ,  CC >. ) )
6 cnex 9467 . . 3  |-  CC  e.  _V
7 op1stg 6692 . . 3  |-  ( ( A  e.  ( -u pi (,] pi )  /\  CC  e.  _V )  -> 
( 1st `  <. A ,  CC >. )  =  A )
86, 7mpan2 671 . 2  |-  ( A  e.  ( -u pi (,] pi )  ->  ( 1st `  <. A ,  CC >. )  =  A )
95, 8eqtrd 2492 1  |-  ( A  e.  ( -u pi (,] pi )  ->  ( 1st `  (inftyexpi  `  A ) )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3071   <.cop 3984   ` cfv 5519  (class class class)co 6193   1stc1st 6678   CCcc 9384   -ucneg 9700   (,]cioc 11405   picpi 13463  inftyexpi cinftyexpi 32838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fv 5527  df-1st 6680  df-bj-inftyexpi 32839
This theorem is referenced by:  bj-inftyexpiinj  32841
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