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Theorem cnneiima 48814
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 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
cnneiima.2 (𝜑𝑁 ∈ ((nei‘𝐾)‘𝑇))
cnneiima.3 (𝜑𝑆 ⊆ (𝐹𝑇))
Assertion
Ref Expression
cnneiima (𝜑 → (𝐹𝑁) ∈ ((nei‘𝐽)‘𝑆))

Proof of Theorem cnneiima
StepHypRef Expression
1 cnneiima.3 . . . 4 (𝜑𝑆 ⊆ (𝐹𝑇))
2 cnneiima.1 . . . . . . 7 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
3 eqid 2737 . . . . . . . 8 𝐽 = 𝐽
4 eqid 2737 . . . . . . . 8 𝐾 = 𝐾
53, 4cnf 23254 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
62, 5syl 17 . . . . . 6 (𝜑𝐹: 𝐽 𝐾)
76ffund 6740 . . . . 5 (𝜑 → Fun 𝐹)
8 cnneiima.2 . . . . . 6 (𝜑𝑁 ∈ ((nei‘𝐾)‘𝑇))
9 cntop2 23249 . . . . . . . 8 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
102, 9syl 17 . . . . . . 7 (𝜑𝐾 ∈ Top)
114neiss2 23109 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐾)‘𝑇)) → 𝑇 𝐾)
1210, 8, 11syl2anc 584 . . . . . . 7 (𝜑𝑇 𝐾)
134neii1 23114 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐾)‘𝑇)) → 𝑁 𝐾)
1410, 8, 13syl2anc 584 . . . . . . 7 (𝜑𝑁 𝐾)
154neiint 23112 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝑇 𝐾𝑁 𝐾) → (𝑁 ∈ ((nei‘𝐾)‘𝑇) ↔ 𝑇 ⊆ ((int‘𝐾)‘𝑁)))
1610, 12, 14, 15syl3anc 1373 . . . . . 6 (𝜑 → (𝑁 ∈ ((nei‘𝐾)‘𝑇) ↔ 𝑇 ⊆ ((int‘𝐾)‘𝑁)))
178, 16mpbid 232 . . . . 5 (𝜑𝑇 ⊆ ((int‘𝐾)‘𝑁))
18 sspreima 7088 . . . . 5 ((Fun 𝐹𝑇 ⊆ ((int‘𝐾)‘𝑁)) → (𝐹𝑇) ⊆ (𝐹 “ ((int‘𝐾)‘𝑁)))
197, 17, 18syl2anc 584 . . . 4 (𝜑 → (𝐹𝑇) ⊆ (𝐹 “ ((int‘𝐾)‘𝑁)))
201, 19sstrd 3994 . . 3 (𝜑𝑆 ⊆ (𝐹 “ ((int‘𝐾)‘𝑁)))
214cnntri 23279 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑁 𝐾) → (𝐹 “ ((int‘𝐾)‘𝑁)) ⊆ ((int‘𝐽)‘(𝐹𝑁)))
222, 14, 21syl2anc 584 . . 3 (𝜑 → (𝐹 “ ((int‘𝐾)‘𝑁)) ⊆ ((int‘𝐽)‘(𝐹𝑁)))
2320, 22sstrd 3994 . 2 (𝜑𝑆 ⊆ ((int‘𝐽)‘(𝐹𝑁)))
24 cntop1 23248 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
252, 24syl 17 . . 3 (𝜑𝐽 ∈ Top)
26 sspreima 7088 . . . . . 6 ((Fun 𝐹𝑇 𝐾) → (𝐹𝑇) ⊆ (𝐹 𝐾))
277, 12, 26syl2anc 584 . . . . 5 (𝜑 → (𝐹𝑇) ⊆ (𝐹 𝐾))
28 fimacnv 6758 . . . . . 6 (𝐹: 𝐽 𝐾 → (𝐹 𝐾) = 𝐽)
296, 28syl 17 . . . . 5 (𝜑 → (𝐹 𝐾) = 𝐽)
3027, 29sseqtrd 4020 . . . 4 (𝜑 → (𝐹𝑇) ⊆ 𝐽)
311, 30sstrd 3994 . . 3 (𝜑𝑆 𝐽)
32 sspreima 7088 . . . . 5 ((Fun 𝐹𝑁 𝐾) → (𝐹𝑁) ⊆ (𝐹 𝐾))
337, 14, 32syl2anc 584 . . . 4 (𝜑 → (𝐹𝑁) ⊆ (𝐹 𝐾))
3433, 29sseqtrd 4020 . . 3 (𝜑 → (𝐹𝑁) ⊆ 𝐽)
353neiint 23112 . . 3 ((𝐽 ∈ Top ∧ 𝑆 𝐽 ∧ (𝐹𝑁) ⊆ 𝐽) → ((𝐹𝑁) ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘(𝐹𝑁))))
3625, 31, 34, 35syl3anc 1373 . 2 (𝜑 → ((𝐹𝑁) ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘(𝐹𝑁))))
3723, 36mpbird 257 1 (𝜑 → (𝐹𝑁) ∈ ((nei‘𝐽)‘𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  wss 3951   cuni 4907  ccnv 5684  cima 5688  Fun wfun 6555  wf 6557  cfv 6561  (class class class)co 7431  Topctop 22899  intcnt 23025  neicnei 23105   Cn ccn 23232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-top 22900  df-topon 22917  df-ntr 23028  df-nei 23106  df-cn 23235
This theorem is referenced by:  sepfsepc  48825
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