Theorem bj-inftyexpiinv 33095

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.

Step	Hyp	Ref	Expression
1		opeq1 4402	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, ℂ⟩ = ⟨𝐴, ℂ⟩)
2		df-bj-inftyexpi 33094	. . . 4 ⊢ inftyexpi = (𝑥 ∈ (-π(,]π) ↦ ⟨𝑥, ℂ⟩)
3		opex 4932	. . . 4 ⊢ ⟨𝐴, ℂ⟩ ∈ V
4	1, 2, 3	fvmpt 6282	. . 3 ⊢ (𝐴 ∈ (-π(,]π) → (inftyexpi ‘𝐴) = ⟨𝐴, ℂ⟩)
5	4	fveq2d 6195	. 2 ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘(inftyexpi ‘𝐴)) = (1st ‘⟨𝐴, ℂ⟩))
6		cnex 10017	. . 3 ⊢ ℂ ∈ V
7		op1stg 7180	. . 3 ⊢ ((𝐴 ∈ (-π(,]π) ∧ ℂ ∈ V) → (1st ‘⟨𝐴, ℂ⟩) = 𝐴)
8	6, 7	mpan2 707	. 2 ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘⟨𝐴, ℂ⟩) = 𝐴)
9	5, 8	eqtrd 2656	1 ⊢ (𝐴 ∈ (-π(,]π) → (1st ‘(inftyexpi ‘𝐴)) = 𝐴)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  ℂcc 9934  -cneg 10267  (,]cioc 12176  πcpi 14797  inftyexpi cinftyexpi 33093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-bj-inftyexpi 33094

This theorem is referenced by:  bj-inftyexpiinj  33096