Theorem cnss2 14950

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 2231	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
2		cnss2.1	. . . . . 6 ⊢ 𝑌 = ∪ 𝐾
3	1, 2	cnf 14927	. . . . 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 529	. . . . 5 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿 ⊆ 𝐾)
6		cnima 14943	. . . . . . 7 ⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥 ∈ 𝐾) → (◡𝑓 “ 𝑥) ∈ 𝐽)
7	6	ralrimiva 2605	. . . . . 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 3291	. . . . 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 14924	. . . . . . 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 14741	. . . . . 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 527	. . . . 5 ⊢ (((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝐿 ∈ (TopOn‘𝑌))
16		iscn 14920	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝐿 ∈ (TopOn‘𝑌)) → (𝑓 ∈ (𝐽 Cn 𝐿) ↔ (𝑓:∪ 𝐽⟶𝑌 ∧ ∀𝑥 ∈ 𝐿 (◡𝑓 “ 𝑥) ∈ 𝐽)))
17	14, 15, 16	syl2anc 411	. . . 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 952	. . 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 3233	1 ⊢ ((𝐿 ∈ (TopOn‘𝑌) ∧ 𝐿 ⊆ 𝐾) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn 𝐿))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  ∀wral 2510   ⊆ wss 3200  ∪ cuni 3893  ^◡ccnv 4724   “ cima 4728  ⟶wf 5322  ‘cfv 5326  (class class class)co 6017  Topctop 14720  TopOnctopon 14733   Cn ccn 14908

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-top 14721  df-topon 14734  df-cn 14911

This theorem is referenced by: (None)