Theorem topssnei 23333

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 1211	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐾 ∈ Top)
2		simprl 783	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐽 ⊆ 𝐾)
3		simpl1 1210	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐽 ∈ Top)
4		simprr 785	. . . . . . . 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 23315	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑥 ⊆ 𝑋)
7	3, 4, 6	syl2anc 596	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ⊆ 𝑋)
8	5	ntropn 23258	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
9	3, 7, 8	syl2anc 596	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
10	2, 9	sseldd 3939	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ 𝐾)
11	5	neiss2 23310	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)
12	3, 4, 11	syl2anc 596	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑆 ⊆ 𝑋)
13	5	neiint 23313	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑥)))
14	3, 12, 7, 13	syl3anc 1398	. . . . . 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 23327	. . . . 5 ⊢ ((𝐾 ∈ Top ∧ ((int‘𝐽)‘𝑥) ∈ 𝐾 ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑥)) → ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆))
17	1, 10, 15, 16	syl3anc 1398	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆))
18	5	ntrss2 23266	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
19	3, 7, 18	syl2anc 596	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
20		simpl3 1212	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑋 = 𝑌)
21	7, 20	sseqtrd 3974	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ⊆ 𝑌)
22		topssnei.2	. . . . 5 ⊢ 𝑌 = ∪ 𝐾
23	22	ssnei2 23325	. . . 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 852	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ∈ ((nei‘𝐾)‘𝑆))
25	24	expr 462	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥 ∈ ((nei‘𝐾)‘𝑆)))
26	25	ssrdv 3944	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽 ⊆ 𝐾) → ((nei‘𝐽)‘𝑆) ⊆ ((nei‘𝐾)‘𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ⊆ wss 3906  ∪ cuni 4874  ‘cfv 6540  Topctop 23102  intcnt 23226  neicnei 23306

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-top 23103  df-ntr 23229  df-nei 23307

This theorem is used by:  flimss1  24183