Theorem cnneiima 49941

Description: Given a continuous function, the preimage of a neighborhood is a neighborhood. To be precise, the preimage of a neighborhood of a subset 𝑇 of the codomain of a continuous function is a neighborhood of any subset of the preimage of 𝑇. (Contributed by Zhi Wang, 9-Sep-2024.)

Hypotheses

Ref	Expression
cnneiima.1	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
cnneiima.2	⊢ (𝜑 → 𝑁 ∈ ((nei‘𝐾)‘𝑇))
cnneiima.3	⊢ (𝜑 → 𝑆 ⊆ (◡𝐹 “ 𝑇))

Assertion

Ref	Expression
cnneiima	⊢ (𝜑 → (◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆))

Proof of Theorem cnneiima

Step	Hyp	Ref	Expression
1		cnneiima.3	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ (◡𝐹 “ 𝑇))
2		cnneiima.1	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
3		eqid 2760	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2760	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 23511	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6	2, 5	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶∪ 𝐾)
7	6	ffund 6703	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
8		cnneiima.2	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ((nei‘𝐾)‘𝑇))
9		cntop2 23506	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
10	2, 9	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
11	4	neiss2 23366	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐾)‘𝑇)) → 𝑇 ⊆ ∪ 𝐾)
12	10, 8, 11	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐾)
13	4	neii1 23371	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐾)‘𝑇)) → 𝑁 ⊆ ∪ 𝐾)
14	10, 8, 13	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → 𝑁 ⊆ ∪ 𝐾)
15	4	neiint 23369	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝑇 ⊆ ∪ 𝐾 ∧ 𝑁 ⊆ ∪ 𝐾) → (𝑁 ∈ ((nei‘𝐾)‘𝑇) ↔ 𝑇 ⊆ ((int‘𝐾)‘𝑁)))
16	10, 12, 14, 15	syl3anc 1398	. . . . . 6 ⊢ (𝜑 → (𝑁 ∈ ((nei‘𝐾)‘𝑇) ↔ 𝑇 ⊆ ((int‘𝐾)‘𝑁)))
17	8, 16	mpbid 235	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ ((int‘𝐾)‘𝑁))
18		sspreima 7056	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑇 ⊆ ((int‘𝐾)‘𝑁)) → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ((int‘𝐾)‘𝑁)))
19	7, 17, 18	syl2anc 596	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ((int‘𝐾)‘𝑁)))
20	1, 19	sstrd 3941	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (◡𝐹 “ ((int‘𝐾)‘𝑁)))
21	4	cnntri 23536	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑁 ⊆ ∪ 𝐾) → (◡𝐹 “ ((int‘𝐾)‘𝑁)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁)))
22	2, 14, 21	syl2anc 596	. . 3 ⊢ (𝜑 → (◡𝐹 “ ((int‘𝐾)‘𝑁)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁)))
23	20, 22	sstrd 3941	. 2 ⊢ (𝜑 → 𝑆 ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁)))
24		cntop1 23505	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
25	2, 24	syl 18	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
26		sspreima 7056	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑇 ⊆ ∪ 𝐾) → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ∪ 𝐾))
27	7, 12, 26	syl2anc 596	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ∪ 𝐾))
28		fimacnv 6721	. . . . . 6 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → (◡𝐹 “ ∪ 𝐾) = ∪ 𝐽)
29	6, 28	syl 18	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ ∪ 𝐾) = ∪ 𝐽)
30	27, 29	sseqtrd 3967	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑇) ⊆ ∪ 𝐽)
31	1, 30	sstrd 3941	. . 3 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)
32		sspreima 7056	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑁 ⊆ ∪ 𝐾) → (◡𝐹 “ 𝑁) ⊆ (◡𝐹 “ ∪ 𝐾))
33	7, 14, 32	syl2anc 596	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑁) ⊆ (◡𝐹 “ ∪ 𝐾))
34	33, 29	sseqtrd 3967	. . 3 ⊢ (𝜑 → (◡𝐹 “ 𝑁) ⊆ ∪ 𝐽)
35	3	neiint 23369	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ (◡𝐹 “ 𝑁) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁))))
36	25, 31, 34, 35	syl3anc 1398	. 2 ⊢ (𝜑 → ((◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁))))
37	23, 36	mpbird 260	1 ⊢ (𝜑 → (◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  ∪ cuni 4867  ^◡ccnv 5647   “ cima 5651  Fun wfun 6522  ⟶wf 6524  ‘cfv 6528  (class class class)co 7409  Topctop 23158  intcnt 23282  neicnei 23362   Cn ccn 23489

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8828  df-top 23159  df-topon 23176  df-ntr 23285  df-nei 23363  df-cn 23492

This theorem is used by:  sepfsepc  49952