Theorem bj-inftyexpitaufo 37706

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 5436	. . . 4 ⊢ ⟨({R‘(1st ‘𝑥)), {R}⟩ ∈ V
2		df-bj-inftyexpitau 37703	. . . 4 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑥)), {R}⟩)
3	1, 2	fnmpti 6668	. . 3 ⊢ +∞eiτ Fn ℝ
4		dffn4 6788	. . 3 ⊢ (+∞eiτ Fn ℝ ↔ +∞eiτ:ℝ–onto→ran +∞eiτ)
5	3, 4	mpbi 233	. 2 ⊢ +∞eiτ:ℝ–onto→ran +∞eiτ
6		df-bj-ccinftyN 37705	. . . 4 ⊢ ℂ∞N = ran +∞eiτ
7	6	eqcomi 2774	. . 3 ⊢ ran +∞eiτ = ℂ∞N
8		foeq3 6780	. . 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 233	1 ⊢ +∞eiτ:ℝ–onto→ℂ∞N

Syntax hints:   ↔ wb 209   = wceq 1563  {csn 4585  ⟨cop 4591  ran crn 5653   Fn wfn 6520  –onto→wfo 6523  ‘cfv 6525  1^st c1st 7972  Rcnr 10838  ℝcr 11087  {_Rcfractemp 37700  +∞e^iτcinftyexpitau 37702  ℂ_∞NcccinftyN 37704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-fun 6527  df-fn 6528  df-fo 6531  df-bj-inftyexpitau 37703  df-bj-ccinftyN 37705

This theorem is referenced by: (None)