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Theorem bj-inftyexpiinj 33595
Description: Injectivity of the parameterization inftyexpi. Remark: a more conceptual proof would use bj-inftyexpiinv 33594 and the fact that a function with a retraction is injective. (Contributed by BJ, 22-Jun-2019.)
Assertion
Ref Expression
bj-inftyexpiinj ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = 𝐵 ↔ (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵)))

Proof of Theorem bj-inftyexpiinj
StepHypRef Expression
1 fveq2 6411 . 2 (𝐴 = 𝐵 → (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵))
2 fveq2 6411 . . 3 ((inftyexpi ‘𝐴) = (inftyexpi ‘𝐵) → (1st ‘(inftyexpi ‘𝐴)) = (1st ‘(inftyexpi ‘𝐵)))
3 bj-inftyexpiinv 33594 . . . . . . 7 (𝐴 ∈ (-π(,]π) → (1st ‘(inftyexpi ‘𝐴)) = 𝐴)
43adantr 473 . . . . . 6 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(inftyexpi ‘𝐴)) = 𝐴)
54eqeq1d 2801 . . . . 5 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(inftyexpi ‘𝐴)) = (1st ‘(inftyexpi ‘𝐵)) ↔ 𝐴 = (1st ‘(inftyexpi ‘𝐵))))
65biimpd 221 . . . 4 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(inftyexpi ‘𝐴)) = (1st ‘(inftyexpi ‘𝐵)) → 𝐴 = (1st ‘(inftyexpi ‘𝐵))))
7 bj-inftyexpiinv 33594 . . . . . 6 (𝐵 ∈ (-π(,]π) → (1st ‘(inftyexpi ‘𝐵)) = 𝐵)
87adantl 474 . . . . 5 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (1st ‘(inftyexpi ‘𝐵)) = 𝐵)
98eqeq2d 2809 . . . 4 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = (1st ‘(inftyexpi ‘𝐵)) ↔ 𝐴 = 𝐵))
106, 9sylibd 231 . . 3 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((1st ‘(inftyexpi ‘𝐴)) = (1st ‘(inftyexpi ‘𝐵)) → 𝐴 = 𝐵))
112, 10syl5 34 . 2 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → ((inftyexpi ‘𝐴) = (inftyexpi ‘𝐵) → 𝐴 = 𝐵))
121, 11impbid2 218 1 ((𝐴 ∈ (-π(,]π) ∧ 𝐵 ∈ (-π(,]π)) → (𝐴 = 𝐵 ↔ (inftyexpi ‘𝐴) = (inftyexpi ‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  cfv 6101  (class class class)co 6878  1st c1st 7399  -cneg 10557  (,]cioc 12425  πcpi 15133  inftyexpi cinftyexpi 33592
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fv 6109  df-1st 7401  df-bj-inftyexpi 33593
This theorem is referenced by:  bj-pinftynminfty  33613
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