Theorem bj-inftyexpitaufo 37187

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 5432	. . . 4 ⊢ ⟨({R‘(1st ‘𝑥)), {R}⟩ ∈ V
2		df-bj-inftyexpitau 37184	. . . 4 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑥)), {R}⟩)
3	1, 2	fnmpti 6669	. . 3 ⊢ +∞eiτ Fn ℝ
4		dffn4 6785	. . 3 ⊢ (+∞eiτ Fn ℝ ↔ +∞eiτ:ℝ–onto→ran +∞eiτ)
5	3, 4	mpbi 230	. 2 ⊢ +∞eiτ:ℝ–onto→ran +∞eiτ
6		df-bj-ccinftyN 37186	. . . 4 ⊢ ℂ∞N = ran +∞eiτ
7	6	eqcomi 2739	. . 3 ⊢ ran +∞eiτ = ℂ∞N
8		foeq3 6777	. . 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 1540  {csn 4597  ⟨cop 4603  ran crn 5647   Fn wfn 6514  –onto→wfo 6517  ‘cfv 6519  1^st c1st 7975  Rcnr 10836  ℝcr 11085  {_Rcfractemp 37181  +∞e^iτcinftyexpitau 37183  ℂ_∞NcccinftyN 37185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-fun 6521  df-fn 6522  df-fo 6525  df-bj-inftyexpitau 37184  df-bj-ccinftyN 37186

This theorem is referenced by: (None)