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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 45602 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 7135 | . 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 24918 | . . . . 5 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ) | |
| 6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ ℂ) | 
| 7 | 3, 6 | sstrd 3994 | . . 3 ⊢ (𝜑 → 𝐷 ⊆ ℂ) | 
| 8 | cncfmptssg.4 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
| 9 | 8 | sselda 3983 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐴) | 
| 10 | cncfmptssg.2 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐸) | |
| 11 | 10 | fvmpt2 7027 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐸 ∈ 𝐷) → (𝐹‘𝑥) = 𝐸) | 
| 12 | 9, 1, 11 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐹‘𝑥) = 𝐸) | 
| 13 | 12 | mpteq2dva 5242 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) = (𝑥 ∈ 𝐶 ↦ 𝐸)) | 
| 14 | nfmpt1 5250 | . . . . . 6 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐸) | |
| 15 | 10, 14 | nfcxfr 2903 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | 
| 16 | 15, 4, 8 | cncfmptss 45602 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)) ∈ (𝐶–cn→𝐵)) | 
| 17 | 13, 16 | eqeltrrd 2842 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐵)) | 
| 18 | cncfcdm 24924 | . . 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 257 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↦ 𝐸) ∈ (𝐶–cn→𝐷)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 –cn→ccncf 24902 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-2 12329 df-cj 15138 df-re 15139 df-im 15140 df-abs 15275 df-cncf 24904 | 
| This theorem is referenced by: negcncfg 45896 itgsinexplem1 45969 itgiccshift 45995 itgperiod 45996 itgsbtaddcnst 45997 dirkeritg 46117 dirkercncflem2 46119 dirkercncflem4 46121 fourierdlem18 46140 fourierdlem23 46145 fourierdlem39 46161 fourierdlem40 46162 fourierdlem62 46183 fourierdlem73 46194 fourierdlem78 46199 fourierdlem83 46204 fourierdlem84 46205 fourierdlem93 46214 fourierdlem95 46216 fourierdlem101 46222 fourierdlem111 46232 etransclem46 46295 | 
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