Mathbox for BJ |
< Previous
Next >
Nearby theorems |
||
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 5337 | . . . 4 ⊢ 〈({R‘(1st ‘𝑥)), {R}〉 ∈ V | |
2 | df-bj-inftyexpitau 35062 | . . . 4 ⊢ +∞eiτ = (𝑥 ∈ ℝ ↦ 〈({R‘(1st ‘𝑥)), {R}〉) | |
3 | 1, 2 | fnmpti 6510 | . . 3 ⊢ +∞eiτ Fn ℝ |
4 | dffn4 6628 | . . 3 ⊢ (+∞eiτ Fn ℝ ↔ +∞eiτ:ℝ–onto→ran +∞eiτ) | |
5 | 3, 4 | mpbi 233 | . 2 ⊢ +∞eiτ:ℝ–onto→ran +∞eiτ |
6 | df-bj-ccinftyN 35064 | . . . 4 ⊢ ℂ∞N = ran +∞eiτ | |
7 | 6 | eqcomi 2743 | . . 3 ⊢ ran +∞eiτ = ℂ∞N |
8 | foeq3 6620 | . . 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 233 | 1 ⊢ +∞eiτ:ℝ–onto→ℂ∞N |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 {csn 4531 〈cop 4537 ran crn 5541 Fn wfn 6364 –onto→wfo 6367 ‘cfv 6369 1st c1st 7748 Rcnr 10462 ℝcr 10711 {Rcfractemp 35059 +∞eiτcinftyexpitau 35061 ℂ∞NcccinftyN 35063 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ral 3059 df-rab 3063 df-v 3403 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-fun 6371 df-fn 6372 df-fo 6375 df-bj-inftyexpitau 35062 df-bj-ccinftyN 35064 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |