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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-inftyexpitaufo | Structured version Visualization version GIF version |
Description: The function +∞eiτ written as a surjection with domain and range. (Contributed by BJ, 4-Feb-2023.) |
Ref | Expression |
---|---|
bj-inftyexpitaufo | ⊢ +∞eiτ:ℝ–onto→ℂ∞N |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5487 | . . . 4 ⊢ 〈({R‘(1st ‘𝑥)), {R}〉 ∈ V | |
2 | df-bj-inftyexpitau 37114 | . . . 4 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ 〈({R‘(1st ‘𝑥)), {R}〉) | |
3 | 1, 2 | fnmpti 6722 | . . 3 ⊢ +∞eiτ Fn ℝ |
4 | dffn4 6839 | . . 3 ⊢ (+∞eiτ Fn ℝ ↔ +∞eiτ:ℝ–onto→ran +∞eiτ) | |
5 | 3, 4 | mpbi 230 | . 2 ⊢ +∞eiτ:ℝ–onto→ran +∞eiτ |
6 | df-bj-ccinftyN 37116 | . . . 4 ⊢ ℂ∞N = ran +∞eiτ | |
7 | 6 | eqcomi 2743 | . . 3 ⊢ ran +∞eiτ = ℂ∞N |
8 | foeq3 6831 | . . 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 230 | 1 ⊢ +∞eiτ: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 –onto→wfo 6570 ‘cfv 6572 1st c1st 8024 Rcnr 10930 ℝcr 11179 {Rcfractemp 37111 +∞eiτcinftyexpitau 37113 ℂ∞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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