Theorem topssnei 23250

Description: A finer topology has more neighborhoods. (Contributed by Mario Carneiro, 9-Apr-2015.)

Hypotheses

Ref	Expression
tpnei.1	⊢ 𝑋 = ∪ 𝐽
topssnei.2	⊢ 𝑌 = ∪ 𝐾

Assertion

Ref	Expression
topssnei	⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽 ⊆ 𝐾) → ((nei‘𝐽)‘𝑆) ⊆ ((nei‘𝐾)‘𝑆))

Proof of Theorem topssnei

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1209	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐾 ∈ Top)
2		simprl 782	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐽 ⊆ 𝐾)
3		simpl1 1208	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐽 ∈ Top)
4		simprr 784	. . . . . . . 8 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ∈ ((nei‘𝐽)‘𝑆))
5		tpnei.1	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
6	5	neii1 23232	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑥 ⊆ 𝑋)
7	3, 4, 6	syl2anc 595	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ⊆ 𝑋)
8	5	ntropn 23175	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
9	3, 7, 8	syl2anc 595	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
10	2, 9	sseldd 3946	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ 𝐾)
11	5	neiss2 23227	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)
12	3, 4, 11	syl2anc 595	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑆 ⊆ 𝑋)
13	5	neiint 23230	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑥)))
14	3, 12, 7, 13	syl3anc 1396	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑥)))
15	4, 14	mpbid 235	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑆 ⊆ ((int‘𝐽)‘𝑥))
16		opnneiss 23244	. . . . 5 ⊢ ((𝐾 ∈ Top ∧ ((int‘𝐽)‘𝑥) ∈ 𝐾 ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑥)) → ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆))
17	1, 10, 15, 16	syl3anc 1396	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆))
18	5	ntrss2 23183	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
19	3, 7, 18	syl2anc 595	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
20		simpl3 1210	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑋 = 𝑌)
21	7, 20	sseqtrd 3981	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ⊆ 𝑌)
22		topssnei.2	. . . . 5 ⊢ 𝑌 = ∪ 𝐾
23	22	ssnei2 23242	. . . 4 ⊢ (((𝐾 ∈ Top ∧ ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆)) ∧ (((int‘𝐽)‘𝑥) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝑌)) → 𝑥 ∈ ((nei‘𝐾)‘𝑆))
24	1, 17, 19, 21, 23	syl22anc 851	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ∈ ((nei‘𝐾)‘𝑆))
25	24	expr 461	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥 ∈ ((nei‘𝐾)‘𝑆)))
26	25	ssrdv 3951	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽 ⊆ 𝐾) → ((nei‘𝐽)‘𝑆) ⊆ ((nei‘𝐾)‘𝑆))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ⊆ wss 3913  ∪ cuni 4876  ‘cfv 6537  Topctop 23019  intcnt 23143  neicnei 23223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-top 23020  df-ntr 23146  df-nei 23224

This theorem is referenced by:  flimss1  24099