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Theorem cnss2 23003
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 π‘Œ = βˆͺ 𝐾
Assertion
Ref Expression
cnss2 ((𝐿 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 βŠ† 𝐾) β†’ (𝐽 Cn 𝐾) βŠ† (𝐽 Cn 𝐿))
Proof of Theorem cnss2
Dummy variables π‘₯ 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2730 . . . . . 6 βˆͺ 𝐽 = βˆͺ 𝐽
2 cnss2.1 . . . . . 6 π‘Œ = βˆͺ 𝐾
31, 2cnf 22972 . . . . 5 (𝑓 ∈ (𝐽 Cn 𝐾) β†’ 𝑓:βˆͺ π½βŸΆπ‘Œ)
43adantl 480 . . . 4 (((𝐿 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 βŠ† 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) β†’ 𝑓:βˆͺ π½βŸΆπ‘Œ)
5 simplr 765 . . . . 5 (((𝐿 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 βŠ† 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) β†’ 𝐿 βŠ† 𝐾)
6 cnima 22991 . . . . . . 7 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ π‘₯ ∈ 𝐾) β†’ (◑𝑓 β€œ π‘₯) ∈ 𝐽)
76ralrimiva 3144 . . . . . 6 (𝑓 ∈ (𝐽 Cn 𝐾) β†’ βˆ€π‘₯ ∈ 𝐾 (◑𝑓 β€œ π‘₯) ∈ 𝐽)
87adantl 480 . . . . 5 (((𝐿 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 βŠ† 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) β†’ βˆ€π‘₯ ∈ 𝐾 (◑𝑓 β€œ π‘₯) ∈ 𝐽)
9 ssralv 4051 . . . . 5 (𝐿 βŠ† 𝐾 β†’ (βˆ€π‘₯ ∈ 𝐾 (◑𝑓 β€œ π‘₯) ∈ 𝐽 β†’ βˆ€π‘₯ ∈ 𝐿 (◑𝑓 β€œ π‘₯) ∈ 𝐽))
105, 8, 9sylc 65 . . . 4 (((𝐿 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 βŠ† 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) β†’ βˆ€π‘₯ ∈ 𝐿 (◑𝑓 β€œ π‘₯) ∈ 𝐽)
11 cntop1 22966 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐾) β†’ 𝐽 ∈ Top)
1211adantl 480 . . . . . 6 (((𝐿 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 βŠ† 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) β†’ 𝐽 ∈ Top)
13 toptopon2 22642 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
1412, 13sylib 217 . . . . 5 (((𝐿 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 βŠ† 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
15 simpll 763 . . . . 5 (((𝐿 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 βŠ† 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) β†’ 𝐿 ∈ (TopOnβ€˜π‘Œ))
16 iscn 22961 . . . . 5 ((𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ 𝐿 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓:βˆͺ π½βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝐿 (◑𝑓 β€œ π‘₯) ∈ 𝐽)))
1714, 15, 16syl2anc 582 . . . 4 (((𝐿 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 βŠ† 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) β†’ (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓:βˆͺ π½βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝐿 (◑𝑓 β€œ π‘₯) ∈ 𝐽)))
184, 10, 17mpbir2and 709 . . 3 (((𝐿 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 βŠ† 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) β†’ 𝑓 ∈ (𝐽 Cn 𝐿))
1918ex 411 . 2 ((𝐿 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 βŠ† 𝐾) β†’ (𝑓 ∈ (𝐽 Cn 𝐾) β†’ 𝑓 ∈ (𝐽 Cn 𝐿)))
2019ssrdv 3989 1 ((𝐿 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐿 βŠ† 𝐾) β†’ (𝐽 Cn 𝐾) βŠ† (𝐽 Cn 𝐿))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059   βŠ† wss 3949  βˆͺ cuni 4909  β—‘ccnv 5676   β€œ cima 5680  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7413  Topctop 22617  TopOnctopon 22634   Cn ccn 22950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7416  df-oprab 7417  df-mpo 7418  df-map 8826  df-top 22618  df-topon 22635  df-cn 22953
This theorem is referenced by:  kgencn3  23284  xmetdcn  24576
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