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Theorem bj-inftyexpitaufo 37170
Description: The function +∞e 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 5484 . . . 4 ⟨({R‘(1st𝑥)), {R}⟩ ∈ V
2 df-bj-inftyexpitau 37167 . . . 4 +∞e = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st𝑥)), {R}⟩)
31, 2fnmpti 6725 . . 3 +∞e Fn ℝ
4 dffn4 6842 . . 3 (+∞e Fn ℝ ↔ +∞e:ℝ–onto→ran +∞e)
53, 4mpbi 230 . 2 +∞e:ℝ–onto→ran +∞e
6 df-bj-ccinftyN 37169 . . . 4 ∞N = ran +∞e
76eqcomi 2749 . . 3 ran +∞e = ℂ∞N
8 foeq3 6834 . . 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 230 1 +∞e:ℝ–onto→ℂ∞N
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  {csn 4648  cop 4654  ran crn 5701   Fn wfn 6570  ontowfo 6573  cfv 6575  1st c1st 8030  Rcnr 10936  cr 11185  {Rcfractemp 37164  +∞ecinftyexpitau 37166  ∞NcccinftyN 37168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-fun 6577  df-fn 6578  df-fo 6581  df-bj-inftyexpitau 37167  df-bj-ccinftyN 37169
This theorem is referenced by: (None)
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