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Theorem cnss2 20891
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 2610 . . . . . 6 𝐽 = 𝐽
2 cnss2.1 . . . . . 6 𝑌 = 𝐾
31, 2cnf 20860 . . . . 5 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓: 𝐽𝑌)
43adantl 481 . . . 4 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓: 𝐽𝑌)
5 simplr 788 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿𝐾)
6 cnima 20879 . . . . . . 7 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥𝐾) → (𝑓𝑥) ∈ 𝐽)
76ralrimiva 2949 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) → ∀𝑥𝐾 (𝑓𝑥) ∈ 𝐽)
87adantl 481 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥𝐾 (𝑓𝑥) ∈ 𝐽)
9 ssralv 3629 . . . . 5 (𝐿𝐾 → (∀𝑥𝐾 (𝑓𝑥) ∈ 𝐽 → ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽))
105, 8, 9sylc 63 . . . 4 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽)
11 cntop1 20854 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
1211adantl 481 . . . . . 6 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
131toptopon 20548 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1412, 13sylib 207 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘ 𝐽))
15 simpll 786 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿 ∈ (TopOn‘𝑌))
16 iscn 20849 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓: 𝐽𝑌 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽)))
1714, 15, 16syl2anc 691 . . . 4 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓: 𝐽𝑌 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽)))
184, 10, 17mpbir2and 959 . . 3 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐿))
1918ex 449 . 2 ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ (𝐽 Cn 𝐿)))
2019ssrdv 3574 1 ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wss 3540   cuni 4372  ccnv 5037  cima 5041  wf 5800  cfv 5804  (class class class)co 6549  Topctop 20517  TopOnctopon 20518   Cn ccn 20838
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
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  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-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-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-top 20521  df-topon 20523  df-cn 20841
This theorem is referenced by:  kgencn3  21171  xmetdcn  22449
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