Theorem cnneiima 49183

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 2736	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
4		eqid 2736	. . . . . . . 8 ⊢ ∪ 𝐾 = ∪ 𝐾
5	3, 4	cnf 23192	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾)
6	2, 5	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶∪ 𝐾)
7	6	ffund 6666	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
8		cnneiima.2	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ((nei‘𝐾)‘𝑇))
9		cntop2 23187	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
10	2, 9	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Top)
11	4	neiss2 23047	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐾)‘𝑇)) → 𝑇 ⊆ ∪ 𝐾)
12	10, 8, 11	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ ∪ 𝐾)
13	4	neii1 23052	. . . . . . . 8 ⊢ ((𝐾 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐾)‘𝑇)) → 𝑁 ⊆ ∪ 𝐾)
14	10, 8, 13	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑁 ⊆ ∪ 𝐾)
15	4	neiint 23050	. . . . . . 7 ⊢ ((𝐾 ∈ Top ∧ 𝑇 ⊆ ∪ 𝐾 ∧ 𝑁 ⊆ ∪ 𝐾) → (𝑁 ∈ ((nei‘𝐾)‘𝑇) ↔ 𝑇 ⊆ ((int‘𝐾)‘𝑁)))
16	10, 12, 14, 15	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → (𝑁 ∈ ((nei‘𝐾)‘𝑇) ↔ 𝑇 ⊆ ((int‘𝐾)‘𝑁)))
17	8, 16	mpbid 232	. . . . 5 ⊢ (𝜑 → 𝑇 ⊆ ((int‘𝐾)‘𝑁))
18		sspreima 7013	. . . . 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 3944	. . 3 ⊢ (𝜑 → 𝑆 ⊆ (◡𝐹 “ ((int‘𝐾)‘𝑁)))
21	4	cnntri 23217	. . . 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 3944	. 2 ⊢ (𝜑 → 𝑆 ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁)))
24		cntop1 23186	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
25	2, 24	syl 17	. . 3 ⊢ (𝜑 → 𝐽 ∈ Top)
26		sspreima 7013	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑇 ⊆ ∪ 𝐾) → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ∪ 𝐾))
27	7, 12, 26	syl2anc 584	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ 𝑇) ⊆ (◡𝐹 “ ∪ 𝐾))
28		fimacnv 6684	. . . . . 6 ⊢ (𝐹:∪ 𝐽⟶∪ 𝐾 → (◡𝐹 “ ∪ 𝐾) = ∪ 𝐽)
29	6, 28	syl 17	. . . . 5 ⊢ (𝜑 → (◡𝐹 “ ∪ 𝐾) = ∪ 𝐽)
30	27, 29	sseqtrd 3970	. . . 4 ⊢ (𝜑 → (◡𝐹 “ 𝑇) ⊆ ∪ 𝐽)
31	1, 30	sstrd 3944	. . 3 ⊢ (𝜑 → 𝑆 ⊆ ∪ 𝐽)
32		sspreima 7013	. . . . 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 3970	. . 3 ⊢ (𝜑 → (◡𝐹 “ 𝑁) ⊆ ∪ 𝐽)
35	3	neiint 23050	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ (◡𝐹 “ 𝑁) ⊆ ∪ 𝐽) → ((◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁))))
36	25, 31, 34, 35	syl3anc 1373	. 2 ⊢ (𝜑 → ((◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘(◡𝐹 “ 𝑁))))
37	23, 36	mpbird 257	1 ⊢ (𝜑 → (◡𝐹 “ 𝑁) ∈ ((nei‘𝐽)‘𝑆))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113   ⊆ wss 3901  ∪ cuni 4863  ^◡ccnv 5623   “ cima 5627  Fun wfun 6486  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  Topctop 22839  intcnt 22963  neicnei 23043   Cn ccn 23170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8767  df-top 22840  df-topon 22857  df-ntr 22966  df-nei 23044  df-cn 23173

This theorem is referenced by:  sepfsepc  49194