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Mirrors > Home > MPE Home > Th. List > Mathboxes > cncfmptssg | Structured version Visualization version GIF version |
Description: A continuous complex function restricted to a subset is continuous, using maps-to notation. This theorem generalizes cncfmptss 41744 because it allows to establish a subset for the codomain also. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
cncfmptssg.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐸) |
cncfmptssg.3 | ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵)) |
cncfmptssg.4 | ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
cncfmptssg.5 | ⊢ (𝜑 → 𝐷 ⊆ 𝐵) |
cncfmptssg.6 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐸 ∈ 𝐷) |
Ref | Expression |
---|---|
cncfmptssg | ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncfmptssg.6 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐸 ∈ 𝐷) | |
2 | 1 | fmpttd 6871 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐸):𝐶⟶𝐷) |
3 | cncfmptssg.5 | . . . 4 ⊢ (𝜑 → 𝐷 ⊆ 𝐵) | |
4 | cncfmptssg.3 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵)) | |
5 | cncfrss2 23427 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ) |
7 | 3, 6 | sstrd 3974 | . . 3 ⊢ (𝜑 → 𝐷 ⊆ ℂ) |
8 | cncfmptssg.4 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
9 | 8 | sselda 3964 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐴) |
10 | cncfmptssg.2 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐸) | |
11 | 10 | fvmpt2 6771 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐸 ∈ 𝐷) → (𝐹‘𝑥) = 𝐸) |
12 | 9, 1, 11 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐹‘𝑥) = 𝐸) |
13 | 12 | mpteq2dva 5152 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐶 ↦ 𝐸)) |
14 | nfmpt1 5155 | . . . . . 6 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐸) | |
15 | 10, 14 | nfcxfr 2972 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
16 | 15, 4, 8 | cncfmptss 41744 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵)) |
17 | 13, 16 | eqeltrrd 2911 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐵)) |
18 | cncffvrn 23433 | . . 3 ⊢ ((𝐷 ⊆ ℂ ∧ (𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐵)) → ((𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐷) ↔ (𝑥 ∈ 𝐶 ↦ 𝐸):𝐶⟶𝐷)) | |
19 | 7, 17, 18 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐷) ↔ (𝑥 ∈ 𝐶 ↦ 𝐸):𝐶⟶𝐷)) |
20 | 2, 19 | mpbird 258 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 ↦ cmpt 5137 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 –cn→ccncf 23411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-2 11688 df-cj 14446 df-re 14447 df-im 14448 df-abs 14583 df-cncf 23413 |
This theorem is referenced by: negcncfg 42040 itgsinexplem1 42115 itgiccshift 42141 itgperiod 42142 itgsbtaddcnst 42143 dirkeritg 42264 dirkercncflem2 42266 dirkercncflem4 42268 fourierdlem18 42287 fourierdlem23 42292 fourierdlem39 42308 fourierdlem40 42309 fourierdlem62 42330 fourierdlem73 42341 fourierdlem78 42346 fourierdlem83 42351 fourierdlem84 42352 fourierdlem93 42361 fourierdlem95 42363 fourierdlem101 42369 fourierdlem111 42379 etransclem46 42442 |
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