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Theorem bj-inftyexpitaufo 34617
 Description: The function +∞eiτ written as a surjection with domain and range. (Contributed by BJ, 4-Feb-2023.)
Assertion
Ref Expression
bj-inftyexpitaufo +∞e:ℝ–onto→ℂ∞N

Proof of Theorem bj-inftyexpitaufo
StepHypRef Expression
1 opex 5321 . . . 4 ⟨({R‘(1st𝑥)), {R}⟩ ∈ V
2 df-bj-inftyexpitau 34614 . . . 4 +∞e = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st𝑥)), {R}⟩)
31, 2fnmpti 6463 . . 3 +∞e Fn ℝ
4 dffn4 6571 . . 3 (+∞e Fn ℝ ↔ +∞e:ℝ–onto→ran +∞e)
53, 4mpbi 233 . 2 +∞e:ℝ–onto→ran +∞e
6 df-bj-ccinftyN 34616 . . . 4 ∞N = ran +∞e
76eqcomi 2807 . . 3 ran +∞e = ℂ∞N
8 foeq3 6563 . . 3 (ran +∞e = ℂ∞N → (+∞e:ℝ–onto→ran +∞e ↔ +∞e:ℝ–onto→ℂ∞N))
97, 8ax-mp 5 . 2 (+∞e:ℝ–onto→ran +∞e ↔ +∞e:ℝ–onto→ℂ∞N)
105, 9mpbi 233 1 +∞e:ℝ–onto→ℂ∞N
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538  {csn 4525  ⟨cop 4531  ran crn 5520   Fn wfn 6319  –onto→wfo 6322  ‘cfv 6324  1st c1st 7669  Rcnr 10276  ℝcr 10525  {Rcfractemp 34611  +∞eiτcinftyexpitau 34613  ℂ∞NcccinftyN 34615 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-fun 6326  df-fn 6327  df-fo 6330  df-bj-inftyexpitau 34614  df-bj-ccinftyN 34616 This theorem is referenced by: (None)
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