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Theorem cnss2 21301
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 2770 . . . . . 6 𝐽 = 𝐽
2 cnss2.1 . . . . . 6 𝑌 = 𝐾
31, 2cnf 21270 . . . . 5 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓: 𝐽𝑌)
43adantl 467 . . . 4 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓: 𝐽𝑌)
5 simplr 744 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿𝐾)
6 cnima 21289 . . . . . . 7 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥𝐾) → (𝑓𝑥) ∈ 𝐽)
76ralrimiva 3114 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) → ∀𝑥𝐾 (𝑓𝑥) ∈ 𝐽)
87adantl 467 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥𝐾 (𝑓𝑥) ∈ 𝐽)
9 ssralv 3813 . . . . 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 21264 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
1211adantl 467 . . . . . 6 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
131toptopon 20941 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1412, 13sylib 208 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘ 𝐽))
15 simpll 742 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿 ∈ (TopOn‘𝑌))
16 iscn 21259 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓: 𝐽𝑌 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽)))
1714, 15, 16syl2anc 565 . . . 4 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓: 𝐽𝑌 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽)))
184, 10, 17mpbir2and 684 . . 3 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐿))
1918ex 397 . 2 ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ (𝐽 Cn 𝐿)))
2019ssrdv 3756 1 ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wral 3060  wss 3721   cuni 4572  ccnv 5248  cima 5252  wf 6027  cfv 6031  (class class class)co 6792  Topctop 20917  TopOnctopon 20934   Cn ccn 21248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-map 8010  df-top 20918  df-topon 20935  df-cn 21251
This theorem is referenced by:  kgencn3  21581  xmetdcn  22860
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