Theorem bj-inftyexpitaufo 35988

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 5460	. . . 4 ⊢ ⟨({R‘(1st ‘𝑥)), {R}⟩ ∈ V
2		df-bj-inftyexpitau 35985	. . . 4 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st ‘𝑥)), {R}⟩)
3	1, 2	fnmpti 6683	. . 3 ⊢ +∞eiτ Fn ℝ
4		dffn4 6801	. . 3 ⊢ (+∞eiτ Fn ℝ ↔ +∞eiτ:ℝ–onto→ran +∞eiτ)
5	3, 4	mpbi 229	. 2 ⊢ +∞eiτ:ℝ–onto→ran +∞eiτ
6		df-bj-ccinftyN 35987	. . . 4 ⊢ ℂ∞N = ran +∞eiτ
7	6	eqcomi 2742	. . 3 ⊢ ran +∞eiτ = ℂ∞N
8		foeq3 6793	. . 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 1542  {csn 4624  ⟨cop 4630  ran crn 5673   Fn wfn 6530  –onto→wfo 6533  ‘cfv 6535  1^st c1st 7960  Rcnr 10847  ℝcr 11096  {_Rcfractemp 35982  +∞e^iτcinftyexpitau 35984  ℂ_∞NcccinftyN 35986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-fun 6537  df-fn 6538  df-fo 6541  df-bj-inftyexpitau 35985  df-bj-ccinftyN 35987

This theorem is referenced by: (None)