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Theorem bj-inftyexpiinv 33432
 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 ∈ (-π(,]π) → (1st ‘(inftyexpi ‘𝐴)) = 𝐴)

Proof of Theorem bj-inftyexpiinv
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 opeq1 4539 . . . 4 (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2 df-bj-inftyexpi 33431 . . . 4 inftyexpi = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
3 opex 5060 . . . 4 𝐴, ℂ⟩ ∈ V
41, 2, 3fvmpt 6424 . . 3 (𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩)
54fveq2d 6336 . 2 (𝐴 ∈ (-π(,]π) → (1st ‘(inftyexpi ‘𝐴)) = (1st ‘⟨𝐴, ℂ⟩))
6 cnex 10219 . . 3 ℂ ∈ V
7 op1stg 7327 . . 3 ((𝐴 ∈ (-π(,]π) ∧ ℂ ∈ V) → (1st ‘⟨𝐴, ℂ⟩) = 𝐴)
86, 7mpan2 671 . 2 (𝐴 ∈ (-π(,]π) → (1st ‘⟨𝐴, ℂ⟩) = 𝐴)
95, 8eqtrd 2805 1 (𝐴 ∈ (-π(,]π) → (1st ‘(inftyexpi ‘𝐴)) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145  Vcvv 3351  ⟨cop 4322  ‘cfv 6031  (class class class)co 6793  1st c1st 7313  ℂcc 10136  -cneg 10469  (,]cioc 12381  πcpi 15003  inftyexpi cinftyexpi 33430 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-1st 7315  df-bj-inftyexpi 33431 This theorem is referenced by:  bj-inftyexpiinj  33433
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