Theorem bj-inftyexpitaufo 37235

Description: The function +∞e^iτ written as a surjection with domain and range. (Contributed by BJ, 4-Feb-2023.)

Assertion

Ref	Expression
bj-inftyexpitaufo	⊢ +∞eiτ:ℝ–onto→ℂ∞N

Proof of Theorem bj-inftyexpitaufo

Step	Hyp	Ref	Expression
1		opex 5404	. . . 4 ⊢ ⟨({R‘(1st ‘𝑥)), {R}⟩ ∈ V
2		df-bj-inftyexpitau 37232	. . . 4 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑥)), {R}⟩)
3	1, 2	fnmpti 6624	. . 3 ⊢ +∞eiτ Fn ℝ
4		dffn4 6741	. . 3 ⊢ (+∞eiτ Fn ℝ ↔ +∞eiτ:ℝ–onto→ran +∞eiτ)
5	3, 4	mpbi 230	. 2 ⊢ +∞eiτ:ℝ–onto→ran +∞eiτ
6		df-bj-ccinftyN 37234	. . . 4 ⊢ ℂ∞N = ran +∞eiτ
7	6	eqcomi 2740	. . 3 ⊢ ran +∞eiτ = ℂ∞N
8		foeq3 6733	. . 3 ⊢ (ran +∞eiτ = ℂ∞N → (+∞eiτ:ℝ–onto→ran +∞eiτ ↔ +∞eiτ:ℝ–onto→ℂ∞N))
9	7, 8	ax-mp 5	. 2 ⊢ (+∞eiτ:ℝ–onto→ran +∞eiτ ↔ +∞eiτ:ℝ–onto→ℂ∞N)
10	5, 9	mpbi 230	1 ⊢ +∞eiτ:ℝ–onto→ℂ∞N

Syntax hints:   ↔ wb 206   = wceq 1541  {csn 4576  ⟨cop 4582  ran crn 5617   Fn wfn 6476  –onto→wfo 6479  ‘cfv 6481  1^st c1st 7919  Rcnr 10753  ℝcr 11002  {_Rcfractemp 37229  +∞e^iτcinftyexpitau 37231  ℂ_∞NcccinftyN 37233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-fun 6483  df-fn 6484  df-fo 6487  df-bj-inftyexpitau 37232  df-bj-ccinftyN 37234

This theorem is referenced by: (None)