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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 43918 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 7067 | . 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 24278 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ) |
7 | 3, 6 | sstrd 3958 | . . 3 ⊢ (𝜑 → 𝐷 ⊆ ℂ) |
8 | cncfmptssg.4 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
9 | 8 | sselda 3948 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐴) |
10 | cncfmptssg.2 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐸) | |
11 | 10 | fvmpt2 6963 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐸 ∈ 𝐷) → (𝐹‘𝑥) = 𝐸) |
12 | 9, 1, 11 | syl2anc 585 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐹‘𝑥) = 𝐸) |
13 | 12 | mpteq2dva 5209 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐶 ↦ 𝐸)) |
14 | nfmpt1 5217 | . . . . . 6 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐸) | |
15 | 10, 14 | nfcxfr 2902 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
16 | 15, 4, 8 | cncfmptss 43918 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵)) |
17 | 13, 16 | eqeltrrd 2835 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐵)) |
18 | cncfcdm 24284 | . . 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 585 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐷) ↔ (𝑥 ∈ 𝐶 ↦ 𝐸):𝐶⟶𝐷)) |
20 | 2, 19 | mpbird 257 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3914 ↦ cmpt 5192 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 ℂcc 11057 –cn→ccncf 24262 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-2 12224 df-cj 14993 df-re 14994 df-im 14995 df-abs 15130 df-cncf 24264 |
This theorem is referenced by: negcncfg 44212 itgsinexplem1 44285 itgiccshift 44311 itgperiod 44312 itgsbtaddcnst 44313 dirkeritg 44433 dirkercncflem2 44435 dirkercncflem4 44437 fourierdlem18 44456 fourierdlem23 44461 fourierdlem39 44477 fourierdlem40 44478 fourierdlem62 44499 fourierdlem73 44510 fourierdlem78 44515 fourierdlem83 44520 fourierdlem84 44521 fourierdlem93 44530 fourierdlem95 44532 fourierdlem101 44538 fourierdlem111 44548 etransclem46 44611 |
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