Theorem bj-inftyexpitaufo 38315

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⟶ontoℂ∞N

Proof of Theorem bj-inftyexpitaufo

Step	Hyp	Ref	Expression
1		opex 5436	. . . 4 ⊢ ⟨({R‘(1st ‘𝑥)), {R}⟩ ∈ V
2		df-bj-inftyexpitau 38312	. . . 4 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑥)), {R}⟩)
3	1, 2	fnmpti 6704	. . 3 ⊢ +∞eiτ Fn ℝ
4		dffn4 6824	. . 3 ⊢ (+∞eiτ Fn ℝ ↔ +∞eiτ:ℝ⟶ontoran +∞eiτ)
5	3, 4	mpbi 233	. 2 ⊢ +∞eiτ:ℝ⟶ontoran +∞eiτ
6		df-bj-ccinftyN 38314	. . . 4 ⊢ ℂ∞N = ran +∞eiτ
7	6	eqcomi 2772	. . 3 ⊢ ran +∞eiτ = ℂ∞N
8		foeq3 6816	. . 3 ⊢ (ran +∞eiτ = ℂ∞N → (+∞eiτ:ℝ⟶ontoran +∞eiτ ↔ +∞eiτ:ℝ⟶ontoℂ∞N))
9	7, 8	ax-mp 5	. 2 ⊢ (+∞eiτ:ℝ⟶ontoran +∞eiτ ↔ +∞eiτ:ℝ⟶ontoℂ∞N)
10	5, 9	mpbi 233	1 ⊢ +∞eiτ:ℝ⟶ontoℂ∞N

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570  {csn 4588  ⟨cop 4838  ran crn 5656   Fn wfn 6553  ⟶ontowfo 6556  ‘cfv 6558  1^st c1st 8023  Rcnr 10977  ℝcr 11226  {_Rcfractemp 38309  +∞e^iτcinftyexpitau 38311  ℂ_∞NcccinftyN 38313

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-fun 6560  df-fn 6561  df-fo 6564  df-bj-inftyexpitau 38312  df-bj-ccinftyN 38314

This theorem is used by: (None)