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Theorem cnss1 23224
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 2725 . . . . . 6 𝐿 = 𝐿
31, 2cnf 23194 . . . . 5 (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓:𝑋 𝐿)
43adantl 480 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓:𝑋 𝐿)
5 simpllr 774 . . . . . 6 ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥𝐿) → 𝐽𝐾)
6 cnima 23213 . . . . . . 7 ((𝑓 ∈ (𝐽 Cn 𝐿) ∧ 𝑥𝐿) → (𝑓𝑥) ∈ 𝐽)
76adantll 712 . . . . . 6 ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥𝐿) → (𝑓𝑥) ∈ 𝐽)
85, 7sseldd 3977 . . . . 5 ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥𝐿) → (𝑓𝑥) ∈ 𝐾)
98ralrimiva 3135 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐾)
10 simpll 765 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐾 ∈ (TopOn‘𝑋))
11 cntop2 23189 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
1211adantl 480 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ Top)
13 toptopon2 22864 . . . . . 6 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
1412, 13sylib 217 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ (TopOn‘ 𝐿))
15 iscn 23183 . . . . 5 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘ 𝐿)) → (𝑓 ∈ (𝐾 Cn 𝐿) ↔ (𝑓:𝑋 𝐿 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐾)))
1610, 14, 15syl2anc 582 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → (𝑓 ∈ (𝐾 Cn 𝐿) ↔ (𝑓:𝑋 𝐿 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐾)))
174, 9, 16mpbir2and 711 . . 3 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓 ∈ (𝐾 Cn 𝐿))
1817ex 411 . 2 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓 ∈ (𝐾 Cn 𝐿)))
1918ssrdv 3982 1 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wss 3944   cuni 4909  ccnv 5677  cima 5681  wf 6545  cfv 6549  (class class class)co 7419  Topctop 22839  TopOnctopon 22856   Cn ccn 23172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847  df-top 22840  df-topon 22857  df-cn 23175
This theorem is referenced by:  kgen2cn  23507  xkopjcn  23604
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