Theorem cnss1 23594

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 2760	. . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿
3	1, 2	cnf 23564	. . . . 5 ⊢ (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓:𝑋⟶∪ 𝐿)
4	3	adantl 487	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓:𝑋⟶∪ 𝐿)
5		simpllr 788	. . . . . 6 ⊢ ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → 𝐽 ⊆ 𝐾)
6		cnima 23583	. . . . . . 7 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐿) ∧ 𝑥 ∈ 𝐿) → (◡𝑓 “ 𝑥) ∈ 𝐽)
7	6	adantll 727	. . . . . 6 ⊢ ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡𝑓 “ 𝑥) ∈ 𝐽)
8	5, 7	sseldd 3909	. . . . 5 ⊢ ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡𝑓 “ 𝑥) ∈ 𝐾)
9	8	ralrimiva 3154	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐾)
10		simpll 779	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐾 ∈ (TopOn‘𝑋))
11		cntop2 23559	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
12	11	adantl 487	. . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ Top)
13		toptopon2 23236	. . . . . 6 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
14	12, 13	sylib 221	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ (TopOn‘∪ 𝐿))
15		iscn 23553	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿)) → (𝑓 ∈ (𝐾 Cn 𝐿) ↔ (𝑓:𝑋⟶∪ 𝐿 ∧ ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐾)))
16	10, 14, 15	syl2anc 596	. . . 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 726	. . 3 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓 ∈ (𝐾 Cn 𝐿))
18	17	ex 418	. 2 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓 ∈ (𝐾 Cn 𝐿)))
19	18	ssrdv 3914	1 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895  ∪ cuni 4866  ^◡ccnv 5649   “ cima 5653  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  Topctop 23211  TopOnctopon 23228   Cn ccn 23542

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8849  df-top 23212  df-topon 23229  df-cn 23545

This theorem is used by:  kgen2cn  23878  xkopjcn  23975