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Theorem cnss1 13386
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 2177 . . . . . 6 𝐿 = 𝐿
31, 2cnf 13364 . . . . 5 (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓:𝑋 𝐿)
43adantl 277 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓:𝑋 𝐿)
5 simpllr 534 . . . . . 6 ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥𝐿) → 𝐽𝐾)
6 cnima 13380 . . . . . . 7 ((𝑓 ∈ (𝐽 Cn 𝐿) ∧ 𝑥𝐿) → (𝑓𝑥) ∈ 𝐽)
76adantll 476 . . . . . 6 ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥𝐿) → (𝑓𝑥) ∈ 𝐽)
85, 7sseldd 3156 . . . . 5 ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥𝐿) → (𝑓𝑥) ∈ 𝐾)
98ralrimiva 2550 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐾)
10 simpll 527 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐾 ∈ (TopOn‘𝑋))
11 cntop2 13362 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
1211adantl 277 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ Top)
132toptopon 13176 . . . . . 6 (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘ 𝐿))
1412, 13sylib 122 . . . . 5 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ (TopOn‘ 𝐿))
15 iscn 13357 . . . . 5 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘ 𝐿)) → (𝑓 ∈ (𝐾 Cn 𝐿) ↔ (𝑓:𝑋 𝐿 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐾)))
1610, 14, 15syl2anc 411 . . . 4 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → (𝑓 ∈ (𝐾 Cn 𝐿) ↔ (𝑓:𝑋 𝐿 ∧ ∀𝑥𝐿 (𝑓𝑥) ∈ 𝐾)))
174, 9, 16mpbir2and 944 . . 3 (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓 ∈ (𝐾 Cn 𝐿))
1817ex 115 . 2 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓 ∈ (𝐾 Cn 𝐿)))
1918ssrdv 3161 1 ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  wss 3129   cuni 3807  ccnv 4622  cima 4626  wf 5208  cfv 5212  (class class class)co 5869  Topctop 13155  TopOnctopon 13168   Cn ccn 13345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-map 6644  df-top 13156  df-topon 13169  df-cn 13348
This theorem is referenced by: (None)
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