Theorem topssnei 22183

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 1190	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐾 ∈ Top)
2		simprl 767	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐽 ⊆ 𝐾)
3		simpl1 1189	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐽 ∈ Top)
4		simprr 769	. . . . . . . 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 22165	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑥 ⊆ 𝑋)
7	3, 4, 6	syl2anc 583	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ⊆ 𝑋)
8	5	ntropn 22108	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
9	3, 7, 8	syl2anc 583	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
10	2, 9	sseldd 3918	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ 𝐾)
11	5	neiss2 22160	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)
12	3, 4, 11	syl2anc 583	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑆 ⊆ 𝑋)
13	5	neiint 22163	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑥)))
14	3, 12, 7, 13	syl3anc 1369	. . . . . 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 231	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑆 ⊆ ((int‘𝐽)‘𝑥))
16		opnneiss 22177	. . . . 5 ⊢ ((𝐾 ∈ Top ∧ ((int‘𝐽)‘𝑥) ∈ 𝐾 ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑥)) → ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆))
17	1, 10, 15, 16	syl3anc 1369	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆))
18	5	ntrss2 22116	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
19	3, 7, 18	syl2anc 583	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
20		simpl3 1191	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑋 = 𝑌)
21	7, 20	sseqtrd 3957	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ⊆ 𝑌)
22		topssnei.2	. . . . 5 ⊢ 𝑌 = ∪ 𝐾
23	22	ssnei2 22175	. . . 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 835	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ∈ ((nei‘𝐾)‘𝑆))
25	24	expr 456	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥 ∈ ((nei‘𝐾)‘𝑆)))
26	25	ssrdv 3923	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽 ⊆ 𝐾) → ((nei‘𝐽)‘𝑆) ⊆ ((nei‘𝐾)‘𝑆))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  ∪ cuni 4836  ‘cfv 6418  Topctop 21950  intcnt 22076  neicnei 22156

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-top 21951  df-ntr 22079  df-nei 22157

This theorem is referenced by:  flimss1  23032