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Theorem bj-inftyexpitaufo 37117
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 5487 . . . 4 ⟨({R‘(1st𝑥)), {R}⟩ ∈ V
2 df-bj-inftyexpitau 37114 . . . 4 +∞e = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st𝑥)), {R}⟩)
31, 2fnmpti 6722 . . 3 +∞e Fn ℝ
4 dffn4 6839 . . 3 (+∞e Fn ℝ ↔ +∞e:ℝ–onto→ran +∞e)
53, 4mpbi 230 . 2 +∞e:ℝ–onto→ran +∞e
6 df-bj-ccinftyN 37116 . . . 4 ∞N = ran +∞e
76eqcomi 2743 . . 3 ran +∞e = ℂ∞N
8 foeq3 6831 . . 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 5700   Fn wfn 6567  ontowfo 6570  cfv 6572  1st c1st 8024  Rcnr 10930  cr 11179  {Rcfractemp 37111  +∞ecinftyexpitau 37113  ∞NcccinftyN 37115
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 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
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 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-mpt 5253  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-fun 6574  df-fn 6575  df-fo 6578  df-bj-inftyexpitau 37114  df-bj-ccinftyN 37116
This theorem is referenced by: (None)
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