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Theorem bj-inftyexpiinj 34959
Description: Injectivity of the parameterization inftyexpi. Remark: a more conceptual proof would use bj-inftyexpiinv 34958 and the fact that a function with a retraction is injective. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-inftyexpiinj  |-  ( ( A  e.  ( -u pi (,] pi )  /\  B  e.  ( -u pi (,] pi ) )  -> 
( A  =  B  <-> 
(inftyexpi  `  A )  =  (inftyexpi  `  B ) ) )

Proof of Theorem bj-inftyexpiinj
StepHypRef Expression
1 fveq2 5774 . 2  |-  ( A  =  B  ->  (inftyexpi  `  A )  =  (inftyexpi  `  B ) )
2 fveq2 5774 . . 3  |-  ( (inftyexpi  `  A )  =  (inftyexpi  `  B )  ->  ( 1st `  (inftyexpi  `  A ) )  =  ( 1st `  (inftyexpi  `  B ) ) )
3 bj-inftyexpiinv 34958 . . . . . . 7  |-  ( A  e.  ( -u pi (,] pi )  ->  ( 1st `  (inftyexpi  `  A ) )  =  A )
43adantr 463 . . . . . 6  |-  ( ( A  e.  ( -u pi (,] pi )  /\  B  e.  ( -u pi (,] pi ) )  -> 
( 1st `  (inftyexpi  `  A ) )  =  A )
54eqeq1d 2384 . . . . 5  |-  ( ( A  e.  ( -u pi (,] pi )  /\  B  e.  ( -u pi (,] pi ) )  -> 
( ( 1st `  (inftyexpi  `  A ) )  =  ( 1st `  (inftyexpi  `  B ) )  <->  A  =  ( 1st `  (inftyexpi  `  B
) ) ) )
65biimpd 207 . . . 4  |-  ( ( A  e.  ( -u pi (,] pi )  /\  B  e.  ( -u pi (,] pi ) )  -> 
( ( 1st `  (inftyexpi  `  A ) )  =  ( 1st `  (inftyexpi  `  B ) )  ->  A  =  ( 1st `  (inftyexpi  `  B ) ) ) )
7 bj-inftyexpiinv 34958 . . . . . 6  |-  ( B  e.  ( -u pi (,] pi )  ->  ( 1st `  (inftyexpi  `  B ) )  =  B )
87adantl 464 . . . . 5  |-  ( ( A  e.  ( -u pi (,] pi )  /\  B  e.  ( -u pi (,] pi ) )  -> 
( 1st `  (inftyexpi  `  B ) )  =  B )
98eqeq2d 2396 . . . 4  |-  ( ( A  e.  ( -u pi (,] pi )  /\  B  e.  ( -u pi (,] pi ) )  -> 
( A  =  ( 1st `  (inftyexpi  `  B
) )  <->  A  =  B ) )
106, 9sylibd 214 . . 3  |-  ( ( A  e.  ( -u pi (,] pi )  /\  B  e.  ( -u pi (,] pi ) )  -> 
( ( 1st `  (inftyexpi  `  A ) )  =  ( 1st `  (inftyexpi  `  B ) )  ->  A  =  B )
)
112, 10syl5 32 . 2  |-  ( ( A  e.  ( -u pi (,] pi )  /\  B  e.  ( -u pi (,] pi ) )  -> 
( (inftyexpi  `  A )  =  (inftyexpi  `  B )  ->  A  =  B ) )
121, 11impbid2 204 1  |-  ( ( A  e.  ( -u pi (,] pi )  /\  B  e.  ( -u pi (,] pi ) )  -> 
( A  =  B  <-> 
(inftyexpi  `  A )  =  (inftyexpi  `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   ` cfv 5496  (class class class)co 6196   1stc1st 6697   -ucneg 9719   (,]cioc 11451   picpi 13804  inftyexpi cinftyexpi 34956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5460  df-fun 5498  df-fv 5504  df-1st 6699  df-bj-inftyexpi 34957
This theorem is referenced by:  bj-pinftynminfty  34977
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