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Mirrors > Home > MPE Home > Th. List > cncfss | Structured version Visualization version GIF version |
Description: The set of continuous functions is expanded when the codomain is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.) |
Ref | Expression |
---|---|
cncfss | ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ ℂ) → (𝐴–cn→𝐵) ⊆ (𝐴–cn→𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cncff 24272 | . . . . . 6 ⊢ (𝑓 ∈ (𝐴–cn→𝐵) → 𝑓:𝐴⟶𝐵) | |
2 | 1 | adantl 483 | . . . . 5 ⊢ (((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ ℂ) ∧ 𝑓 ∈ (𝐴–cn→𝐵)) → 𝑓:𝐴⟶𝐵) |
3 | simpll 766 | . . . . 5 ⊢ (((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ ℂ) ∧ 𝑓 ∈ (𝐴–cn→𝐵)) → 𝐵 ⊆ 𝐶) | |
4 | 2, 3 | fssd 6691 | . . . 4 ⊢ (((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ ℂ) ∧ 𝑓 ∈ (𝐴–cn→𝐵)) → 𝑓:𝐴⟶𝐶) |
5 | cncfcdm 24277 | . . . . 5 ⊢ ((𝐶 ⊆ ℂ ∧ 𝑓 ∈ (𝐴–cn→𝐵)) → (𝑓 ∈ (𝐴–cn→𝐶) ↔ 𝑓:𝐴⟶𝐶)) | |
6 | 5 | adantll 713 | . . . 4 ⊢ (((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ ℂ) ∧ 𝑓 ∈ (𝐴–cn→𝐵)) → (𝑓 ∈ (𝐴–cn→𝐶) ↔ 𝑓:𝐴⟶𝐶)) |
7 | 4, 6 | mpbird 257 | . . 3 ⊢ (((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ ℂ) ∧ 𝑓 ∈ (𝐴–cn→𝐵)) → 𝑓 ∈ (𝐴–cn→𝐶)) |
8 | 7 | ex 414 | . 2 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ ℂ) → (𝑓 ∈ (𝐴–cn→𝐵) → 𝑓 ∈ (𝐴–cn→𝐶))) |
9 | 8 | ssrdv 3955 | 1 ⊢ ((𝐵 ⊆ 𝐶 ∧ 𝐶 ⊆ ℂ) → (𝐴–cn→𝐵) ⊆ (𝐴–cn→𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3915 ⟶wf 6497 (class class class)co 7362 ℂcc 11056 –cn→ccncf 24255 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-2 12223 df-cj 14991 df-re 14992 df-im 14993 df-abs 15128 df-cncf 24257 |
This theorem is referenced by: cncfcompt2 24287 cncfmptid 24292 cncfmpt2ss 24295 evthicc2 24840 volivth 24987 iblabslem 25208 iblabs 25209 bddmulibl 25219 cnlimci 25269 rolle 25370 c1liplem1 25376 dvivth 25390 dvcnvrelem2 25398 itgsubst 25429 logcn 26018 logccv 26034 fdvposlt 33252 fdvneggt 33253 fdvposle 33254 fdvnegge 33255 logdivsqrle 33303 knoppcnlem10 34994 ftc1cnnclem 36178 ftc2nc 36189 areacirclem2 36196 evthiccabs 43808 cncfcompt 44198 cncficcgt0 44203 cncfiooicc 44209 cncfiooiccre 44210 itgsubsticclem 44290 fourierdlem72 44493 fourierdlem78 44499 fourierdlem83 44504 fourierdlem84 44505 fourierdlem85 44506 fourierdlem88 44509 fourierdlem95 44516 fourierdlem111 44532 |
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