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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 5348 | . . . 4 ⊢ 〈({R‘(1st ‘𝑥)), {R}〉 ∈ V | |
2 | df-bj-inftyexpitau 34475 | . . . 4 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ 〈({R‘(1st ‘𝑥)), {R}〉) | |
3 | 1, 2 | fnmpti 6485 | . . 3 ⊢ +∞eiτ Fn ℝ |
4 | dffn4 6590 | . . 3 ⊢ (+∞eiτ Fn ℝ ↔ +∞eiτ:ℝ–onto→ran +∞eiτ) | |
5 | 3, 4 | mpbi 232 | . 2 ⊢ +∞eiτ:ℝ–onto→ran +∞eiτ |
6 | df-bj-ccinftyN 34477 | . . . 4 ⊢ ℂ∞N = ran +∞eiτ | |
7 | 6 | eqcomi 2830 | . . 3 ⊢ ran +∞eiτ = ℂ∞N |
8 | foeq3 6582 | . . 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 232 | 1 ⊢ +∞eiτ:ℝ–onto→ℂ∞N |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 {csn 4560 〈cop 4566 ran crn 5550 Fn wfn 6344 –onto→wfo 6347 ‘cfv 6349 1st c1st 7681 Rcnr 10281 ℝcr 10530 {Rcfractemp 34472 +∞eiτcinftyexpitau 34474 ℂ∞NcccinftyN 34476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-fun 6351 df-fn 6352 df-fo 6355 df-bj-inftyexpitau 34475 df-bj-ccinftyN 34477 |
This theorem is referenced by: (None) |
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