Theorem cnneiima 49543

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 2764	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2764	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 23308	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6	2, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶∪ 𝐾)
7	6	ffund 6698	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
8		cnneiima.2	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ((nei‘𝐾)‘𝑇))
9		cntop2 23303	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
10	2, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
11	4	neiss2 23163	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐾)‘𝑇)) → 𝑇 ⊆ ∪ 𝐾)
12	10, 8, 11	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐾)
13	4	neii1 23168	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐾)‘𝑇)) → 𝑁 ⊆ ∪ 𝐾)
14	10, 8, 13	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → 𝑁 ⊆ ∪ 𝐾)
15	4	neiint 23166	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝑇 ⊆ ∪ 𝐾 ∧ 𝑁 ⊆ ∪ 𝐾) → (𝑁 ∈ ((nei‘𝐾)‘𝑇) ↔ 𝑇 ⊆ ((int‘𝐾)‘𝑁)))
16	10, 12, 14, 15	syl3anc 1392	. . . . . 6 ⊢ (𝜑 → (𝑁 ∈ ((nei‘𝐾)‘𝑇) ↔ 𝑇 ⊆ ((int‘𝐾)‘𝑁)))
17	8, 16	mpbid 234	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ ((int‘𝐾)‘𝑁))
18		sspreima 7051	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑇 ⊆ ((int‘𝐾)‘𝑁)) → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ((int‘𝐾)‘𝑁)))
19	7, 17, 18	syl2anc 593	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ((int‘𝐾)‘𝑁)))
20	1, 19	sstrd 3948	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (◡𝐹 “ ((int‘𝐾)‘𝑁)))
21	4	cnntri 23333	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑁 ⊆ ∪ 𝐾) → (◡𝐹 “ ((int‘𝐾)‘𝑁)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁)))
22	2, 14, 21	syl2anc 593	. . 3 ⊢ (𝜑 → (◡𝐹 “ ((int‘𝐾)‘𝑁)) ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁)))
23	20, 22	sstrd 3948	. 2 ⊢ (𝜑 → 𝑆 ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁)))
24		cntop1 23302	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
25	2, 24	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
26		sspreima 7051	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑇 ⊆ ∪ 𝐾) → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ∪ 𝐾))
27	7, 12, 26	syl2anc 593	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ∪ 𝐾))
28		fimacnv 6716	. . . . . 6 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → (◡𝐹 “ ∪ 𝐾) = ∪ 𝐽)
29	6, 28	syl 17	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ ∪ 𝐾) = ∪ 𝐽)
30	27, 29	sseqtrd 3974	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑇) ⊆ ∪ 𝐽)
31	1, 30	sstrd 3948	. . 3 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)
32		sspreima 7051	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑁 ⊆ ∪ 𝐾) → (◡𝐹 “ 𝑁) ⊆ (◡𝐹 “ ∪ 𝐾))
33	7, 14, 32	syl2anc 593	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑁) ⊆ (◡𝐹 “ ∪ 𝐾))
34	33, 29	sseqtrd 3974	. . 3 ⊢ (𝜑 → (◡𝐹 “ 𝑁) ⊆ ∪ 𝐽)
35	3	neiint 23166	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ (◡𝐹 “ 𝑁) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁))))
36	25, 31, 34, 35	syl3anc 1392	. 2 ⊢ (𝜑 → ((◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁))))
37	23, 36	mpbird 259	1 ⊢ (𝜑 → (◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1562   ∈ wcel 2144   ⊆ wss 3906  ∪ cuni 4867  ^◡ccnv 5648   “ cima 5652  Fun wfun 6517  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398  Topctop 22955  intcnt 23079  neicnei 23159   Cn ccn 23286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-map 8812  df-top 22956  df-topon 22973  df-ntr 23082  df-nei 23160  df-cn 23289

This theorem is referenced by:  sepfsepc  49554