Theorem topssnei 14830

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 1025	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐾 ∈ Top)
2		simprl 529	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐽 ⊆ 𝐾)
3		simpl1 1024	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐽 ∈ Top)
4		simprr 531	. . . . . . . 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 14815	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑥 ⊆ 𝑋)
7	3, 4, 6	syl2anc 411	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ⊆ 𝑋)
8	5	ntropn 14785	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
9	3, 7, 8	syl2anc 411	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
10	2, 9	sseldd 3225	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ 𝐾)
11	5	neiss2 14810	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)
12	3, 4, 11	syl2anc 411	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑆 ⊆ 𝑋)
13	5	neiint 14813	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑥)))
14	3, 12, 7, 13	syl3anc 1271	. . . . . 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 147	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑆 ⊆ ((int‘𝐽)‘𝑥))
16		opnneiss 14826	. . . . 5 ⊢ ((𝐾 ∈ Top ∧ ((int‘𝐽)‘𝑥) ∈ 𝐾 ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑥)) → ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆))
17	1, 10, 15, 16	syl3anc 1271	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆))
18	5	ntrss2 14789	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
19	3, 7, 18	syl2anc 411	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
20		simpl3 1026	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑋 = 𝑌)
21	7, 20	sseqtrd 3262	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ⊆ 𝑌)
22		topssnei.2	. . . . 5 ⊢ 𝑌 = ∪ 𝐾
23	22	ssnei2 14825	. . . 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 1272	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ∈ ((nei‘𝐾)‘𝑆))
25	24	expr 375	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥 ∈ ((nei‘𝐾)‘𝑆)))
26	25	ssrdv 3230	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽 ⊆ 𝐾) → ((nei‘𝐽)‘𝑆) ⊆ ((nei‘𝐾)‘𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200   ⊆ wss 3197  ∪ cuni 3887  ‘cfv 5317  Topctop 14665  intcnt 14761  neicnei 14806

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-top 14666  df-ntr 14764  df-nei 14807

This theorem is referenced by: (None)