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Theorem cnss1 12454
 Description: If the topology 𝐾 is finer than 𝐽, then there are more continuous functions from 𝐾 than from 𝐽. (Contributed by Mario Carneiro, 19-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)
Hypothesis
Ref Expression
cnss1.1 𝑋 = 𝐽
Assertion
Ref Expression
cnss1 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿))

Proof of Theorem cnss1
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnss1.1 . . . . . 6 𝑋 = 𝐽
2 eqid 2140 . . . . . 6 𝐿 = 𝐿
31, 2cnf 12432 . . . . 5 (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓:𝑋 𝐿)
43adantl 275 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓:𝑋 𝐿)
5 simpllr 524 . . . . . 6 ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥𝐿) → 𝐽𝐾)
6 cnima 12448 . . . . . . 7 ((𝑓 ∈ (𝐽 Cn 𝐿) ∧ 𝑥𝐿) → (𝑓𝑥) ∈ 𝐽)
76adantll 468 . . . . . 6 ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥𝐿) → (𝑓𝑥) ∈ 𝐽)
85, 7sseldd 3104 . . . . 5 ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥𝐿) → (𝑓𝑥) ∈ 𝐾)
98ralrimiva 2509 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐾)
10 simpll 519 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐾 ∈ (TopOn‘𝑋))
11 cntop2 12430 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
1211adantl 275 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ Top)
132toptopon 12244 . . . . . 6 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
1412, 13sylib 121 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ (TopOn‘ 𝐿))
15 iscn 12425 . . . . 5 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘ 𝐿)) → (𝑓 ∈ (𝐾 Cn 𝐿) ↔ (𝑓:𝑋 𝐿 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐾)))
1610, 14, 15syl2anc 409 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → (𝑓 ∈ (𝐾 Cn 𝐿) ↔ (𝑓:𝑋 𝐿 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐾)))
174, 9, 16mpbir2and 929 . . 3 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓 ∈ (𝐾 Cn 𝐿))
1817ex 114 . 2 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓 ∈ (𝐾 Cn 𝐿)))
1918ssrdv 3109 1 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∀wral 2417   ⊆ wss 3077  ∪ cuni 3745  ◡ccnv 4547   “ cima 4551  ⟶wf 5128  ‘cfv 5132  (class class class)co 5783  Topctop 12223  TopOnctopon 12236   Cn ccn 12413 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 4055  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-setind 4461 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 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-fv 5140  df-ov 5786  df-oprab 5787  df-mpo 5788  df-1st 6047  df-2nd 6048  df-map 6553  df-top 12224  df-topon 12237  df-cn 12416 This theorem is referenced by: (None)
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