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Theorem cnpflf2 24008
Description: 𝐹 is continuous at point 𝐴 iff a limit of 𝐹 when 𝑥 tends to 𝐴 is (𝐹𝐴). Proposition 9 of [BourbakiTop1] p. TG I.50. (Contributed by FL, 29-May-2011.) (Revised by Mario Carneiro, 9-Apr-2015.)
Hypothesis
Ref Expression
cnpflf2.3 𝐿 = ((nei‘𝐽)‘{𝐴})
Assertion
Ref Expression
cnpflf2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋𝑌 ∧ (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))))

Proof of Theorem cnpflf2
Dummy variables 𝑢 𝑣 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnpf2 23258 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
213expa 1119 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
323adantl3 1169 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
4 simpl1 1192 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
5 simpl3 1194 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴𝑋)
6 neiflim 23982 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))
7 cnpflf2.3 . . . . . . 7 𝐿 = ((nei‘𝐽)‘{𝐴})
87oveq2i 7442 . . . . . 6 (𝐽 fLim 𝐿) = (𝐽 fLim ((nei‘𝐽)‘{𝐴}))
96, 8eleqtrrdi 2852 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ (𝐽 fLim 𝐿))
104, 5, 9syl2anc 584 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fLim 𝐿))
11 simpr 484 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
12 cnpflfi 24007 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
1310, 11, 12syl2anc 584 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
143, 13jca 511 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹:𝑋𝑌 ∧ (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹)))
15 simpl1 1192 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐽 ∈ (TopOn‘𝑋))
16 topontop 22919 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1715, 16syl 17 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐽 ∈ Top)
18 simpl3 1194 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐴𝑋)
19 toponuni 22920 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2015, 19syl 17 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝑋 = 𝐽)
2118, 20eleqtrd 2843 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐴 𝐽)
227eleq2i 2833 . . . . . . . . . . . 12 (𝑧𝐿𝑧 ∈ ((nei‘𝐽)‘{𝐴}))
23 eqid 2737 . . . . . . . . . . . . 13 𝐽 = 𝐽
2423isneip 23113 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → (𝑧 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑧 𝐽 ∧ ∃𝑣𝐽 (𝐴𝑣𝑣𝑧))))
2522, 24bitrid 283 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → (𝑧𝐿 ↔ (𝑧 𝐽 ∧ ∃𝑣𝐽 (𝐴𝑣𝑣𝑧))))
2617, 21, 25syl2anc 584 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (𝑧𝐿 ↔ (𝑧 𝐽 ∧ ∃𝑣𝐽 (𝐴𝑣𝑣𝑧))))
27 sstr2 3990 . . . . . . . . . . . . . . 15 ((𝐹𝑣) ⊆ (𝐹𝑧) → ((𝐹𝑧) ⊆ 𝑢 → (𝐹𝑣) ⊆ 𝑢))
28 imass2 6120 . . . . . . . . . . . . . . 15 (𝑣𝑧 → (𝐹𝑣) ⊆ (𝐹𝑧))
2927, 28syl11 33 . . . . . . . . . . . . . 14 ((𝐹𝑧) ⊆ 𝑢 → (𝑣𝑧 → (𝐹𝑣) ⊆ 𝑢))
3029anim2d 612 . . . . . . . . . . . . 13 ((𝐹𝑧) ⊆ 𝑢 → ((𝐴𝑣𝑣𝑧) → (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3130reximdv 3170 . . . . . . . . . . . 12 ((𝐹𝑧) ⊆ 𝑢 → (∃𝑣𝐽 (𝐴𝑣𝑣𝑧) → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3231com12 32 . . . . . . . . . . 11 (∃𝑣𝐽 (𝐴𝑣𝑣𝑧) → ((𝐹𝑧) ⊆ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3332adantl 481 . . . . . . . . . 10 ((𝑧 𝐽 ∧ ∃𝑣𝐽 (𝐴𝑣𝑣𝑧)) → ((𝐹𝑧) ⊆ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3426, 33biimtrdi 253 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (𝑧𝐿 → ((𝐹𝑧) ⊆ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))))
3534rexlimdv 3153 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (∃𝑧𝐿 (𝐹𝑧) ⊆ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3635imim2d 57 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (((𝐹𝐴) ∈ 𝑢 → ∃𝑧𝐿 (𝐹𝑧) ⊆ 𝑢) → ((𝐹𝐴) ∈ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))))
3736ralimdv 3169 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑧𝐿 (𝐹𝑧) ⊆ 𝑢) → ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))))
38 simpr 484 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐹:𝑋𝑌)
3937, 38jctild 525 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑧𝐿 (𝐹𝑧) ⊆ 𝑢) → (𝐹:𝑋𝑌 ∧ ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))))
4039adantld 490 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (((𝐹𝐴) ∈ 𝑌 ∧ ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑧𝐿 (𝐹𝑧) ⊆ 𝑢)) → (𝐹:𝑋𝑌 ∧ ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))))
41 simpl2 1193 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐾 ∈ (TopOn‘𝑌))
4218snssd 4809 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → {𝐴} ⊆ 𝑋)
4318snn0d 4775 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → {𝐴} ≠ ∅)
44 neifil 23888 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
4515, 42, 43, 44syl3anc 1373 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
467, 45eqeltrid 2845 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐿 ∈ (Fil‘𝑋))
47 isflf 24001 . . . . 5 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑋𝑌) → ((𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) ↔ ((𝐹𝐴) ∈ 𝑌 ∧ ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑧𝐿 (𝐹𝑧) ⊆ 𝑢))))
4841, 46, 38, 47syl3anc 1373 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → ((𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) ↔ ((𝐹𝐴) ∈ 𝑌 ∧ ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑧𝐿 (𝐹𝑧) ⊆ 𝑢))))
49 iscnp 23245 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))))
5049adantr 480 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))))
5140, 48, 503imtr4d 294 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → ((𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)))
5251impr 454 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ (𝐹:𝑋𝑌 ∧ (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
5314, 52impbida 801 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋𝑌 ∧ (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  wss 3951  c0 4333  {csn 4626   cuni 4907  cima 5688  wf 6557  cfv 6561  (class class class)co 7431  Topctop 22899  TopOnctopon 22916  neicnei 23105   CnP ccnp 23233  Filcfil 23853   fLim cflim 23942   fLimf cflf 23943
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-nel 3047  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-1st 8014  df-2nd 8015  df-map 8868  df-fbas 21361  df-fg 21362  df-top 22900  df-topon 22917  df-ntr 23028  df-nei 23106  df-cnp 23236  df-fil 23854  df-fm 23946  df-flim 23947  df-flf 23948
This theorem is referenced by:  cnpflf  24009
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