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Theorem bj-inftyexpitaufo 33678
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 5164 . . . 4 ⟨({R‘(1st𝑥)), {R}⟩ ∈ V
2 df-bj-inftyexpitau 33675 . . . 4 +∞e = (𝑥 ∈ ℝ ↦ ⟨({R‘(1st𝑥)), {R}⟩)
31, 2fnmpti 6268 . . 3 +∞e Fn ℝ
4 dffn4 6372 . . 3 (+∞e Fn ℝ ↔ +∞e:ℝ–onto→ran +∞e)
53, 4mpbi 222 . 2 +∞e:ℝ–onto→ran +∞e
6 df-bj-ccinftyN 33677 . . . 4 ∞N = ran +∞e
76eqcomi 2787 . . 3 ran +∞e = ℂ∞N
8 foeq3 6364 . . 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 222 1 +∞e:ℝ–onto→ℂ∞N
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1601  {csn 4398  cop 4404  ran crn 5356   Fn wfn 6130  ontowfo 6133  cfv 6135  1st c1st 7443  Rcnr 10022  cr 10271  {Rcfractemp 33672  +∞ecinftyexpitau 33674  ∞NcccinftyN 33676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-fun 6137  df-fn 6138  df-fo 6141  df-bj-inftyexpitau 33675  df-bj-ccinftyN 33677
This theorem is referenced by: (None)
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