Theorem bj-inftyexpitaufo 36751

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 5465	. . . 4 ⊢ ⟨({R‘(1st ‘𝑥)), {R}⟩ ∈ V
2		df-bj-inftyexpitau 36748	. . . 4 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑥)), {R}⟩)
3	1, 2	fnmpti 6697	. . 3 ⊢ +∞eiτ Fn ℝ
4		dffn4 6814	. . 3 ⊢ (+∞eiτ Fn ℝ ↔ +∞eiτ:ℝ–onto→ran +∞eiτ)
5	3, 4	mpbi 229	. 2 ⊢ +∞eiτ:ℝ–onto→ran +∞eiτ
6		df-bj-ccinftyN 36750	. . . 4 ⊢ ℂ∞N = ran +∞eiτ
7	6	eqcomi 2734	. . 3 ⊢ ran +∞eiτ = ℂ∞N
8		foeq3 6806	. . 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 229	1 ⊢ +∞eiτ:ℝ–onto→ℂ∞N

Syntax hints:   ↔ wb 205   = wceq 1533  {csn 4629  ⟨cop 4635  ran crn 5678   Fn wfn 6542  –onto→wfo 6545  ‘cfv 6547  1^st c1st 7990  Rcnr 10888  ℝcr 11137  {_Rcfractemp 36745  +∞e^iτcinftyexpitau 36747  ℂ_∞NcccinftyN 36749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5299  ax-nul 5306  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3948  df-un 3950  df-ss 3962  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-fun 6549  df-fn 6550  df-fo 6553  df-bj-inftyexpitau 36748  df-bj-ccinftyN 36750

This theorem is referenced by: (None)