Theorem cnss1 23263

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 2741	. . . . . 6 ⊢ ∪ 𝐿 = ∪ 𝐿
3	1, 2	cnf 23233	. . . . 5 ⊢ (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓:𝑋⟶∪ 𝐿)
4	3	adantl 483	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓:𝑋⟶∪ 𝐿)
5		simpllr 782	. . . . . 6 ⊢ ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → 𝐽 ⊆ 𝐾)
6		cnima 23252	. . . . . . 7 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐿) ∧ 𝑥 ∈ 𝐿) → (◡𝑓 “ 𝑥) ∈ 𝐽)
7	6	adantll 721	. . . . . 6 ⊢ ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡𝑓 “ 𝑥) ∈ 𝐽)
8	5, 7	sseldd 3918	. . . . 5 ⊢ ((((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) ∧ 𝑥 ∈ 𝐿) → (◡𝑓 “ 𝑥) ∈ 𝐾)
9	8	ralrimiva 3133	. . . 4 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐾)
10		simpll 773	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐾 ∈ (TopOn‘𝑋))
11		cntop2 23228	. . . . . . 7 ⊢ (𝑓 ∈ (𝐽 Cn 𝐿) → 𝐿 ∈ Top)
12	11	adantl 483	. . . . . 6 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ Top)
13		toptopon2 22905	. . . . . 6 ⊢ (𝐿 ∈ Top ↔ 𝐿 ∈ (TopOn‘∪ 𝐿))
14	12, 13	sylib 220	. . . . 5 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝐿 ∈ (TopOn‘∪ 𝐿))
15		iscn 23222	. . . . 5 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (TopOn‘∪ 𝐿)) → (𝑓 ∈ (𝐾 Cn 𝐿) ↔ (𝑓:𝑋⟶∪ 𝐿 ∧ ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐾)))
16	10, 14, 15	syl2anc 591	. . . 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 720	. . 3 ⊢ (((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐿)) → 𝑓 ∈ (𝐾 Cn 𝐿))
18	17	ex 414	. 2 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑓 ∈ (𝐽 Cn 𝐿) → 𝑓 ∈ (𝐾 Cn 𝐿)))
19	18	ssrdv 3923	1 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐽 Cn 𝐿) ⊆ (𝐾 Cn 𝐿))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055   ⊆ wss 3885  ∪ cuni 4841  ^◡ccnv 5620   “ cima 5624  ⟶wf 6485  ‘cfv 6489  (class class class)co 7360  Topctop 22880  TopOnctopon 22897   Cn ccn 23211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8769  df-top 22881  df-topon 22898  df-cn 23214

This theorem is referenced by:  kgen2cn  23546  xkopjcn  23643