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Theorem cnss2 12433
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 2140 . . . . . 6 𝐽 = 𝐽
2 cnss2.1 . . . . . 6 𝑌 = 𝐾
31, 2cnf 12410 . . . . 5 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓: 𝐽𝑌)
43adantl 275 . . . 4 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓: 𝐽𝑌)
5 simplr 520 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿𝐾)
6 cnima 12426 . . . . . . 7 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥𝐾) → (𝑓𝑥) ∈ 𝐽)
76ralrimiva 2508 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) → ∀𝑥𝐾 (𝑓𝑥) ∈ 𝐽)
87adantl 275 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥𝐾 (𝑓𝑥) ∈ 𝐽)
9 ssralv 3165 . . . . 5 (𝐿𝐾 → (∀𝑥𝐾 (𝑓𝑥) ∈ 𝐽 → ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽))
105, 8, 9sylc 62 . . . 4 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽)
11 cntop1 12407 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
1211adantl 275 . . . . . 6 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
131toptopon 12222 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1412, 13sylib 121 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘ 𝐽))
15 simpll 519 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿 ∈ (TopOn‘𝑌))
16 iscn 12403 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓: 𝐽𝑌 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽)))
1714, 15, 16syl2anc 409 . . . 4 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓: 𝐽𝑌 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽)))
184, 10, 17mpbir2and 929 . . 3 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐿))
1918ex 114 . 2 ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ (𝐽 Cn 𝐿)))
2019ssrdv 3107 1 ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wcel 1481  wral 2417  wss 3075   cuni 3743  ccnv 4545  cima 4549  wf 5126  cfv 5130  (class class class)co 5781  Topctop 12201  TopOnctopon 12214   Cn ccn 12391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-map 6551  df-top 12202  df-topon 12215  df-cn 12394
This theorem is referenced by: (None)
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