Theorem topssnei 12703

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 990	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐾 ∈ Top)
2		simprl 521	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐽 ⊆ 𝐾)
3		simpl1 989	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝐽 ∈ Top)
4		simprr 522	. . . . . . . 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 12688	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑥 ⊆ 𝑋)
7	3, 4, 6	syl2anc 409	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ⊆ 𝑋)
8	5	ntropn 12658	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
9	3, 7, 8	syl2anc 409	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ 𝐽)
10	2, 9	sseldd 3138	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ 𝐾)
11	5	neiss2 12683	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋)
12	3, 4, 11	syl2anc 409	. . . . . . 7 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑆 ⊆ 𝑋)
13	5	neiint 12686	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘𝑥)))
14	3, 12, 7, 13	syl3anc 1227	. . . . . 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 146	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑆 ⊆ ((int‘𝐽)‘𝑥))
16		opnneiss 12699	. . . . 5 ⊢ ((𝐾 ∈ Top ∧ ((int‘𝐽)‘𝑥) ∈ 𝐾 ∧ 𝑆 ⊆ ((int‘𝐽)‘𝑥)) → ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆))
17	1, 10, 15, 16	syl3anc 1227	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ∈ ((nei‘𝐾)‘𝑆))
18	5	ntrss2 12662	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ⊆ 𝑋) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
19	3, 7, 18	syl2anc 409	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → ((int‘𝐽)‘𝑥) ⊆ 𝑥)
20		simpl3 991	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑋 = 𝑌)
21	7, 20	sseqtrd 3175	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ⊆ 𝑌)
22		topssnei.2	. . . . 5 ⊢ 𝑌 = ∪ 𝐾
23	22	ssnei2 12698	. . . 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 1228	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ ((nei‘𝐽)‘𝑆))) → 𝑥 ∈ ((nei‘𝐾)‘𝑆))
25	24	expr 373	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ ((nei‘𝐽)‘𝑆) → 𝑥 ∈ ((nei‘𝐾)‘𝑆)))
26	25	ssrdv 3143	1 ⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ 𝑋 = 𝑌) ∧ 𝐽 ⊆ 𝐾) → ((nei‘𝐽)‘𝑆) ⊆ ((nei‘𝐾)‘𝑆))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 967   = wceq 1342   ∈ wcel 2135   ⊆ wss 3111  ∪ cuni 3783  ‘cfv 5182  Topctop 12536  intcnt 12634  neicnei 12679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-top 12537  df-ntr 12637  df-nei 12680

This theorem is referenced by: (None)