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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 "map to" notation. This theorem generalizes cncfmptss 38654 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 | eqid 2610 | . . 3 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐸) = (𝑥 ∈ 𝐶 ↦ 𝐸) | |
3 | 1, 2 | fmptd 6292 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐸):𝐶⟶𝐷) |
4 | cncfmptssg.5 | . . . 4 ⊢ (𝜑 → 𝐷 ⊆ 𝐵) | |
5 | cncfmptssg.3 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐴–cn→𝐵)) | |
6 | cncfrss2 22503 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ) |
8 | 4, 7 | sstrd 3578 | . . 3 ⊢ (𝜑 → 𝐷 ⊆ ℂ) |
9 | cncfmptssg.4 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
10 | 9 | sselda 3568 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐴) |
11 | cncfmptssg.2 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐸) | |
12 | 11 | fvmpt2 6200 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐸 ∈ 𝐷) → (𝐹‘𝑥) = 𝐸) |
13 | 10, 1, 12 | syl2anc 691 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐹‘𝑥) = 𝐸) |
14 | 13 | mpteq2dva 4672 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐶 ↦ 𝐸)) |
15 | nfmpt1 4675 | . . . . . 6 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐸) | |
16 | 11, 15 | nfcxfr 2749 | . . . . 5 ⊢ Ⅎ𝑥𝐹 |
17 | 16, 5, 9 | cncfmptss 38654 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵)) |
18 | 14, 17 | eqeltrrd 2689 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐵)) |
19 | cncffvrn 22509 | . . 3 ⊢ ((𝐷 ⊆ ℂ ∧ (𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐵)) → ((𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐷) ↔ (𝑥 ∈ 𝐶 ↦ 𝐸):𝐶⟶𝐷)) | |
20 | 8, 18, 19 | syl2anc 691 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐷) ↔ (𝑥 ∈ 𝐶 ↦ 𝐸):𝐶⟶𝐷)) |
21 | 3, 20 | mpbird 246 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 –cn→ccncf 22487 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-2 10956 df-cj 13687 df-re 13688 df-im 13689 df-abs 13824 df-cncf 22489 |
This theorem is referenced by: negcncfg 38766 itgsinexplem1 38845 itgiccshift 38872 itgperiod 38873 itgsbtaddcnst 38874 dirkeritg 38995 dirkercncflem2 38997 dirkercncflem4 38999 fourierdlem18 39018 fourierdlem23 39023 fourierdlem39 39039 fourierdlem40 39040 fourierdlem62 39061 fourierdlem73 39072 fourierdlem78 39077 fourierdlem83 39082 fourierdlem84 39083 fourierdlem93 39092 fourierdlem95 39094 fourierdlem101 39100 fourierdlem111 39110 etransclem46 39173 |
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