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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 5164 | . . . 4 ⊢ 〈({R‘(1st ‘𝑥)), {R}〉 ∈ V | |
2 | df-bj-inftyexpitau 33675 | . . . 4 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ 〈({R‘(1st ‘𝑥)), {R}〉) | |
3 | 1, 2 | fnmpti 6268 | . . 3 ⊢ +∞eiτ Fn ℝ |
4 | dffn4 6372 | . . 3 ⊢ (+∞eiτ Fn ℝ ↔ +∞eiτ:ℝ–onto→ran +∞eiτ) | |
5 | 3, 4 | mpbi 222 | . 2 ⊢ +∞eiτ:ℝ–onto→ran +∞eiτ |
6 | df-bj-ccinftyN 33677 | . . . 4 ⊢ ℂ∞N = ran +∞eiτ | |
7 | 6 | eqcomi 2787 | . . 3 ⊢ ran +∞eiτ = ℂ∞N |
8 | foeq3 6364 | . . 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 222 | 1 ⊢ +∞eiτ: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 –onto→wfo 6133 ‘cfv 6135 1st c1st 7443 Rcnr 10022 ℝcr 10271 {Rcfractemp 33672 +∞eiτcinftyexpitau 33674 ℂ∞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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