Theorem cnneiima 46210

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 2738	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2738	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 22397	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6	2, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶∪ 𝐾)
7	6	ffund 6604	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
8		cnneiima.2	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ((nei‘𝐾)‘𝑇))
9		cntop2 22392	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
10	2, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
11	4	neiss2 22252	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐾)‘𝑇)) → 𝑇 ⊆ ∪ 𝐾)
12	10, 8, 11	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐾)
13	4	neii1 22257	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐾)‘𝑇)) → 𝑁 ⊆ ∪ 𝐾)
14	10, 8, 13	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑁 ⊆ ∪ 𝐾)
15	4	neiint 22255	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝑇 ⊆ ∪ 𝐾 ∧ 𝑁 ⊆ ∪ 𝐾) → (𝑁 ∈ ((nei‘𝐾)‘𝑇) ↔ 𝑇 ⊆ ((int‘𝐾)‘𝑁)))
16	10, 12, 14, 15	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → (𝑁 ∈ ((nei‘𝐾)‘𝑇) ↔ 𝑇 ⊆ ((int‘𝐾)‘𝑁)))
17	8, 16	mpbid 231	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ ((int‘𝐾)‘𝑁))
18		sspreima 6945	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑇 ⊆ ((int‘𝐾)‘𝑁)) → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ((int‘𝐾)‘𝑁)))
19	7, 17, 18	syl2anc 584	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ((int‘𝐾)‘𝑁)))
20	1, 19	sstrd 3931	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (◡𝐹 “ ((int‘𝐾)‘𝑁)))
21	4	cnntri 22422	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑁 ⊆ ∪ 𝐾) → (◡𝐹 “ ((int‘𝐾)‘𝑁)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁)))
22	2, 14, 21	syl2anc 584	. . 3 ⊢ (𝜑 → (◡𝐹 “ ((int‘𝐾)‘𝑁)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁)))
23	20, 22	sstrd 3931	. 2 ⊢ (𝜑 → 𝑆 ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁)))
24		cntop1 22391	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
25	2, 24	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
26		sspreima 6945	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑇 ⊆ ∪ 𝐾) → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ∪ 𝐾))
27	7, 12, 26	syl2anc 584	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ∪ 𝐾))
28		fimacnv 6622	. . . . . 6 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → (◡𝐹 “ ∪ 𝐾) = ∪ 𝐽)
29	6, 28	syl 17	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ ∪ 𝐾) = ∪ 𝐽)
30	27, 29	sseqtrd 3961	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑇) ⊆ ∪ 𝐽)
31	1, 30	sstrd 3931	. . 3 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)
32		sspreima 6945	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑁 ⊆ ∪ 𝐾) → (◡𝐹 “ 𝑁) ⊆ (◡𝐹 “ ∪ 𝐾))
33	7, 14, 32	syl2anc 584	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑁) ⊆ (◡𝐹 “ ∪ 𝐾))
34	33, 29	sseqtrd 3961	. . 3 ⊢ (𝜑 → (◡𝐹 “ 𝑁) ⊆ ∪ 𝐽)
35	3	neiint 22255	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ (◡𝐹 “ 𝑁) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁))))
36	25, 31, 34, 35	syl3anc 1370	. 2 ⊢ (𝜑 → ((◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁))))
37	23, 36	mpbird 256	1 ⊢ (𝜑 → (◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887  ∪ cuni 4839  ^◡ccnv 5588   “ cima 5592  Fun wfun 6427  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Topctop 22042  intcnt 22168  neicnei 22248   Cn ccn 22375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-top 22043  df-topon 22060  df-ntr 22171  df-nei 22249  df-cn 22378

This theorem is referenced by:  sepfsepc  46221