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Theorem cnss1 21301
 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 2771 . . . . . 6 𝐿 = 𝐿
31, 2cnf 21271 . . . . 5 (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓:𝑋 𝐿)
43adantl 467 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓:𝑋 𝐿)
5 simpllr 760 . . . . . 6 ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥𝐿) → 𝐽𝐾)
6 cnima 21290 . . . . . . 7 ((𝑓 ∈ (𝐽 Cn 𝐿) ∧ 𝑥𝐿) → (𝑓𝑥) ∈ 𝐽)
76adantll 693 . . . . . 6 ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥𝐿) → (𝑓𝑥) ∈ 𝐽)
85, 7sseldd 3753 . . . . 5 ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥𝐿) → (𝑓𝑥) ∈ 𝐾)
98ralrimiva 3115 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐾)
10 simpll 750 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐾 ∈ (TopOn‘𝑋))
11 cntop2 21266 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
1211adantl 467 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ Top)
132toptopon 20942 . . . . . 6 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
1412, 13sylib 208 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ (TopOn‘ 𝐿))
15 iscn 21260 . . . . 5 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘ 𝐿)) → (𝑓 ∈ (𝐾 Cn 𝐿) ↔ (𝑓:𝑋 𝐿 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐾)))
1610, 14, 15syl2anc 573 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → (𝑓 ∈ (𝐾 Cn 𝐿) ↔ (𝑓:𝑋 𝐿 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐾)))
174, 9, 16mpbir2and 692 . . 3 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓 ∈ (𝐾 Cn 𝐿))
1817ex 397 . 2 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓 ∈ (𝐾 Cn 𝐿)))
1918ssrdv 3758 1 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061   ⊆ wss 3723  ∪ cuni 4575  ◡ccnv 5249   “ cima 5253  ⟶wf 6026  ‘cfv 6030  (class class class)co 6796  Topctop 20918  TopOnctopon 20935   Cn ccn 21249 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-map 8015  df-top 20919  df-topon 20936  df-cn 21252 This theorem is referenced by:  kgen2cn  21583  xkopjcn  21680
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