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Theorem cnneiima 48713
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 2735 . . . . . . . 8 𝐽 = 𝐽
4 eqid 2735 . . . . . . . 8 𝐾 = 𝐾
53, 4cnf 23270 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
62, 5syl 17 . . . . . 6 (𝜑𝐹: 𝐽 𝐾)
76ffund 6741 . . . . 5 (𝜑 → Fun 𝐹)
8 cnneiima.2 . . . . . 6 (𝜑𝑁 ∈ ((nei‘𝐾)‘𝑇))
9 cntop2 23265 . . . . . . . 8 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
102, 9syl 17 . . . . . . 7 (𝜑𝐾 ∈ Top)
114neiss2 23125 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐾)‘𝑇)) → 𝑇 𝐾)
1210, 8, 11syl2anc 584 . . . . . . 7 (𝜑𝑇 𝐾)
134neii1 23130 . . . . . . . 8 ((𝐾 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐾)‘𝑇)) → 𝑁 𝐾)
1410, 8, 13syl2anc 584 . . . . . . 7 (𝜑𝑁 𝐾)
154neiint 23128 . . . . . . 7 ((𝐾 ∈ Top ∧ 𝑇 𝐾𝑁 𝐾) → (𝑁 ∈ ((nei‘𝐾)‘𝑇) ↔ 𝑇 ⊆ ((int‘𝐾)‘𝑁)))
1610, 12, 14, 15syl3anc 1370 . . . . . 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 4006 . . 3 (𝜑𝑆 ⊆ (𝐹 “ ((int‘𝐾)‘𝑁)))
214cnntri 23295 . . . 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 4006 . 2 (𝜑𝑆 ⊆ ((int‘𝐽)‘(𝐹𝑁)))
24 cntop1 23264 . . . 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 6759 . . . . . 6 (𝐹: 𝐽 𝐾 → (𝐹 𝐾) = 𝐽)
296, 28syl 17 . . . . 5 (𝜑 → (𝐹 𝐾) = 𝐽)
3027, 29sseqtrd 4036 . . . 4 (𝜑 → (𝐹𝑇) ⊆ 𝐽)
311, 30sstrd 4006 . . 3 (𝜑𝑆 𝐽)
32 sspreima 7088 . . . . 5 ((Fun 𝐹𝑁 𝐾) → (𝐹𝑁) ⊆ (𝐹 𝐾))
337, 14, 32syl2anc 584 . . . 4 (𝜑 → (𝐹𝑁) ⊆ (𝐹 𝐾))
3433, 29sseqtrd 4036 . . 3 (𝜑 → (𝐹𝑁) ⊆ 𝐽)
353neiint 23128 . . 3 ((𝐽 ∈ Top ∧ 𝑆 𝐽 ∧ (𝐹𝑁) ⊆ 𝐽) → ((𝐹𝑁) ∈ ((nei‘𝐽)‘𝑆) ↔ 𝑆 ⊆ ((int‘𝐽)‘(𝐹𝑁))))
3625, 31, 34, 35syl3anc 1370 . 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 1537  wcel 2106  wss 3963   cuni 4912  ccnv 5688  cima 5692  Fun wfun 6557  wf 6559  cfv 6563  (class class class)co 7431  Topctop 22915  intcnt 23041  neicnei 23121   Cn ccn 23248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-top 22916  df-topon 22933  df-ntr 23044  df-nei 23122  df-cn 23251
This theorem is referenced by:  sepfsepc  48724
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