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Theorem cnss2 22556
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 2738 . . . . . 6 𝐽 = 𝐽
2 cnss2.1 . . . . . 6 𝑌 = 𝐾
31, 2cnf 22525 . . . . 5 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓: 𝐽𝑌)
43adantl 483 . . . 4 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓: 𝐽𝑌)
5 simplr 768 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿𝐾)
6 cnima 22544 . . . . . . 7 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥𝐾) → (𝑓𝑥) ∈ 𝐽)
76ralrimiva 3142 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) → ∀𝑥𝐾 (𝑓𝑥) ∈ 𝐽)
87adantl 483 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥𝐾 (𝑓𝑥) ∈ 𝐽)
9 ssralv 4009 . . . . 5 (𝐿𝐾 → (∀𝑥𝐾 (𝑓𝑥) ∈ 𝐽 → ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽))
105, 8, 9sylc 65 . . . 4 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽)
11 cntop1 22519 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
1211adantl 483 . . . . . 6 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
13 toptopon2 22195 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1412, 13sylib 217 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘ 𝐽))
15 simpll 766 . . . . 5 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿 ∈ (TopOn‘𝑌))
16 iscn 22514 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓: 𝐽𝑌 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽)))
1714, 15, 16syl2anc 585 . . . 4 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓: 𝐽𝑌 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐽)))
184, 10, 17mpbir2and 712 . . 3 (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐿))
1918ex 414 . 2 ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ (𝐽 Cn 𝐿)))
2019ssrdv 3949 1 ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3063  wss 3909   cuni 4864  ccnv 5630  cima 5634  wf 6488  cfv 6492  (class class class)co 7350  Topctop 22170  TopOnctopon 22187   Cn ccn 22503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7353  df-oprab 7354  df-mpo 7355  df-map 8701  df-top 22171  df-topon 22188  df-cn 22506
This theorem is referenced by:  kgencn3  22837  xmetdcn  24129
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