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Theorem cnss2 23285
Description: If the topology 𝐾 is finer than 𝐽, then there are fewer continuous functions into 𝐾 than into 𝐽 from some other space. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
Hypothesis
Ref Expression
cnss2.1 𝑌 = 𝐾
Assertion
Ref Expression
cnss2 ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿))

Proof of Theorem cnss2
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . . 6 𝐽 = 𝐽
2 cnss2.1 . . . . . 6 𝑌 = 𝐾
31, 2cnf 23254 . . . . 5 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓: 𝐽𝑌)
43adantl 481 . . . 4 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓: 𝐽𝑌)
5 simplr 769 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿𝐾)
6 cnima 23273 . . . . . . 7 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥𝐾) → (𝑓𝑥) ∈ 𝐽)
76ralrimiva 3146 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) → ∀𝑥𝐾 (𝑓𝑥) ∈ 𝐽)
87adantl 481 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥𝐾 (𝑓𝑥) ∈ 𝐽)
9 ssralv 4052 . . . . 5 (𝐿𝐾 → (∀𝑥𝐾 (𝑓𝑥) ∈ 𝐽 → ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽))
105, 8, 9sylc 65 . . . 4 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽)
11 cntop1 23248 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
1211adantl 481 . . . . . 6 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
13 toptopon2 22924 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1412, 13sylib 218 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘ 𝐽))
15 simpll 767 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿 ∈ (TopOn‘𝑌))
16 iscn 23243 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓: 𝐽𝑌 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽)))
1714, 15, 16syl2anc 584 . . . 4 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓: 𝐽𝑌 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽)))
184, 10, 17mpbir2and 713 . . 3 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐿))
1918ex 412 . 2 ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ (𝐽 Cn 𝐿)))
2019ssrdv 3989 1 ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wss 3951   cuni 4907  ccnv 5684  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  Topctop 22899  TopOnctopon 22916   Cn ccn 23232
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  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-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-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-top 22900  df-topon 22917  df-cn 23235
This theorem is referenced by:  kgencn3  23566  xmetdcn  24860
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