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Theorem cnss1 23299
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 2734 . . . . . 6 𝐿 = 𝐿
31, 2cnf 23269 . . . . 5 (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓:𝑋 𝐿)
43adantl 481 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓:𝑋 𝐿)
5 simpllr 776 . . . . . 6 ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥𝐿) → 𝐽𝐾)
6 cnima 23288 . . . . . . 7 ((𝑓 ∈ (𝐽 Cn 𝐿) ∧ 𝑥𝐿) → (𝑓𝑥) ∈ 𝐽)
76adantll 714 . . . . . 6 ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥𝐿) → (𝑓𝑥) ∈ 𝐽)
85, 7sseldd 3995 . . . . 5 ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥𝐿) → (𝑓𝑥) ∈ 𝐾)
98ralrimiva 3143 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐾)
10 simpll 767 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐾 ∈ (TopOn‘𝑋))
11 cntop2 23264 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
1211adantl 481 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ Top)
13 toptopon2 22939 . . . . . 6 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
1412, 13sylib 218 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ (TopOn‘ 𝐿))
15 iscn 23258 . . . . 5 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘ 𝐿)) → (𝑓 ∈ (𝐾 Cn 𝐿) ↔ (𝑓:𝑋 𝐿 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐾)))
1610, 14, 15syl2anc 584 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → (𝑓 ∈ (𝐾 Cn 𝐿) ↔ (𝑓:𝑋 𝐿 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐾)))
174, 9, 16mpbir2and 713 . . 3 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓 ∈ (𝐾 Cn 𝐿))
1817ex 412 . 2 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓 ∈ (𝐾 Cn 𝐿)))
1918ssrdv 4000 1 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  wss 3962   cuni 4911  ccnv 5687  cima 5691  wf 6558  cfv 6562  (class class class)co 7430  Topctop 22914  TopOnctopon 22931   Cn ccn 23247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-top 22915  df-topon 22932  df-cn 23250
This theorem is referenced by:  kgen2cn  23582  xkopjcn  23679
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