Theorem cnss1 14900

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) ∧ 𝐽 ⊆ 𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿))

Proof of Theorem cnss1

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

Step	Hyp	Ref	Expression
1		cnss1.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
2		eqid 2229	. . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿
3	1, 2	cnf 14878	. . . . 5 ⊢ (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓:𝑋⟶∪ 𝐿)
4	3	adantl 277	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓:𝑋⟶∪ 𝐿)
5		simpllr 534	. . . . . 6 ⊢ ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → 𝐽 ⊆ 𝐾)
6		cnima 14894	. . . . . . 7 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐿) ∧ 𝑥 ∈ 𝐿) → (◡𝑓 “ 𝑥) ∈ 𝐽)
7	6	adantll 476	. . . . . 6 ⊢ ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡𝑓 “ 𝑥) ∈ 𝐽)
8	5, 7	sseldd 3225	. . . . 5 ⊢ ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡𝑓 “ 𝑥) ∈ 𝐾)
9	8	ralrimiva 2603	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐾)
10		simpll 527	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐾 ∈ (TopOn‘𝑋))
11		cntop2 14876	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
12	11	adantl 277	. . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ Top)
13	2	toptopon 14692	. . . . . 6 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
14	12, 13	sylib 122	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ (TopOn‘∪ 𝐿))
15		iscn 14871	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿)) → (𝑓 ∈ (𝐾 Cn 𝐿) ↔ (𝑓:𝑋⟶∪ 𝐿 ∧ ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐾)))
16	10, 14, 15	syl2anc 411	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → (𝑓 ∈ (𝐾 Cn 𝐿) ↔ (𝑓:𝑋⟶∪ 𝐿 ∧ ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐾)))
17	4, 9, 16	mpbir2and 950	. . 3 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓 ∈ (𝐾 Cn 𝐿))
18	17	ex 115	. 2 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓 ∈ (𝐾 Cn 𝐿)))
19	18	ssrdv 3230	1 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  ∀wral 2508   ⊆ wss 3197  ∪ cuni 3888  ^◡ccnv 4718   “ cima 4722  ⟶wf 5314  ‘cfv 5318  (class class class)co 6001  Topctop 14671  TopOnctopon 14684   Cn ccn 14859

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-map 6797  df-top 14672  df-topon 14685  df-cn 14862

This theorem is referenced by: (None)