Theorem cnss2 15328

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) ∧ 𝐿 ⊆ 𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿))

Proof of Theorem cnss2

Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2238	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
2		cnss2.1	. . . . . 6 ⊢ 𝑌 = ∪ 𝐾
3	1, 2	cnf 15305	. . . . 5 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓:∪ 𝐽⟶𝑌)
4	3	adantl 277	. . . 4 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓:∪ 𝐽⟶𝑌)
5		simplr 533	. . . . 5 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿 ⊆ 𝐾)
6		cnima 15321	. . . . . . 7 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (◡𝑓 “ 𝑥) ∈ 𝐽)
7	6	ralrimiva 2623	. . . . . 6 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → ∀𝑥 ∈ 𝐾 (◡𝑓 “ 𝑥) ∈ 𝐽)
8	7	adantl 277	. . . . 5 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ 𝐾 (◡𝑓 “ 𝑥) ∈ 𝐽)
9		ssralv 3312	. . . . 5 ⊢ (𝐿 ⊆ 𝐾 → (∀𝑥 ∈ 𝐾 (◡𝑓 “ 𝑥) ∈ 𝐽 → ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐽))
10	5, 8, 9	sylc 62	. . . 4 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐽)
11		cntop1 15302	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
12	11	adantl 277	. . . . . 6 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ Top)
13	1	toptopon 15119	. . . . . 6 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
14	12, 13	sylib 122	. . . . 5 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐽 ∈ (TopOn‘∪ 𝐽))
15		simpll 531	. . . . 5 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿 ∈ (TopOn‘𝑌))
16		iscn 15298	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓:∪ 𝐽⟶𝑌 ∧ ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐽)))
17	14, 15, 16	syl2anc 415	. . . 4 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓:∪ 𝐽⟶𝑌 ∧ ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐽)))
18	4, 10, 17	mpbir2and 957	. . 3 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐿))
19	18	ex 115	. 2 ⊢ ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ (𝐽 Cn 𝐿)))
20	19	ssrdv 3254	1 ⊢ ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220  ∪ cuni 3935  ^◡ccnv 4773   “ cima 4777  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085  Topctop 15098  TopOnctopon 15111   Cn ccn 15286

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-top 15099  df-topon 15112  df-cn 15289

This theorem is used by: (None)