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Theorem cnpflf2 22026
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 21277 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
213expa 1112 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
323adantl3 1174 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
4 simpl1 1228 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
5 simpl3 1232 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴𝑋)
6 neiflim 22000 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))
7 cnpflf2.3 . . . . . . 7 𝐿 = ((nei‘𝐽)‘{𝐴})
87oveq2i 6826 . . . . . 6 (𝐽 fLim 𝐿) = (𝐽 fLim ((nei‘𝐽)‘{𝐴}))
96, 8syl6eleqr 2851 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ (𝐽 fLim 𝐿))
104, 5, 9syl2anc 696 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fLim 𝐿))
11 simpr 479 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
12 cnpflfi 22025 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
1310, 11, 12syl2anc 696 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
143, 13jca 555 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹:𝑋𝑌 ∧ (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹)))
15 simpl1 1228 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐽 ∈ (TopOn‘𝑋))
16 topontop 20941 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1715, 16syl 17 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐽 ∈ Top)
18 simpl3 1232 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐴𝑋)
19 toponuni 20942 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2015, 19syl 17 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝑋 = 𝐽)
2118, 20eleqtrd 2842 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐴 𝐽)
227eleq2i 2832 . . . . . . . . . . . 12 (𝑧𝐿𝑧 ∈ ((nei‘𝐽)‘{𝐴}))
23 eqid 2761 . . . . . . . . . . . . 13 𝐽 = 𝐽
2423isneip 21132 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → (𝑧 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑧 𝐽 ∧ ∃𝑣𝐽 (𝐴𝑣𝑣𝑧))))
2522, 24syl5bb 272 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → (𝑧𝐿 ↔ (𝑧 𝐽 ∧ ∃𝑣𝐽 (𝐴𝑣𝑣𝑧))))
2617, 21, 25syl2anc 696 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (𝑧𝐿 ↔ (𝑧 𝐽 ∧ ∃𝑣𝐽 (𝐴𝑣𝑣𝑧))))
27 sstr2 3752 . . . . . . . . . . . . . . 15 ((𝐹𝑣) ⊆ (𝐹𝑧) → ((𝐹𝑧) ⊆ 𝑢 → (𝐹𝑣) ⊆ 𝑢))
28 imass2 5660 . . . . . . . . . . . . . . 15 (𝑣𝑧 → (𝐹𝑣) ⊆ (𝐹𝑧))
2927, 28syl11 33 . . . . . . . . . . . . . 14 ((𝐹𝑧) ⊆ 𝑢 → (𝑣𝑧 → (𝐹𝑣) ⊆ 𝑢))
3029anim2d 590 . . . . . . . . . . . . 13 ((𝐹𝑧) ⊆ 𝑢 → ((𝐴𝑣𝑣𝑧) → (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3130reximdv 3155 . . . . . . . . . . . 12 ((𝐹𝑧) ⊆ 𝑢 → (∃𝑣𝐽 (𝐴𝑣𝑣𝑧) → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3231com12 32 . . . . . . . . . . 11 (∃𝑣𝐽 (𝐴𝑣𝑣𝑧) → ((𝐹𝑧) ⊆ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3332adantl 473 . . . . . . . . . 10 ((𝑧 𝐽 ∧ ∃𝑣𝐽 (𝐴𝑣𝑣𝑧)) → ((𝐹𝑧) ⊆ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3426, 33syl6bi 243 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (𝑧𝐿 → ((𝐹𝑧) ⊆ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))))
3534rexlimdv 3169 . . . . . . . 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 3102 . . . . . 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 479 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐹:𝑋𝑌)
3937, 38jctild 567 . . . . 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 484 . . . 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 1230 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐾 ∈ (TopOn‘𝑌))
4218snssd 4486 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → {𝐴} ⊆ 𝑋)
43 snnzg 4452 . . . . . . . 8 (𝐴𝑋 → {𝐴} ≠ ∅)
4418, 43syl 17 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → {𝐴} ≠ ∅)
45 neifil 21906 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
4615, 42, 44, 45syl3anc 1477 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
477, 46syl5eqel 2844 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐿 ∈ (Fil‘𝑋))
48 isflf 22019 . . . . 5 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑋𝑌) → ((𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) ↔ ((𝐹𝐴) ∈ 𝑌 ∧ ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑧𝐿 (𝐹𝑧) ⊆ 𝑢))))
4941, 47, 38, 48syl3anc 1477 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → ((𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) ↔ ((𝐹𝐴) ∈ 𝑌 ∧ ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑧𝐿 (𝐹𝑧) ⊆ 𝑢))))
50 iscnp 21264 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))))
5150adantr 472 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))))
5240, 49, 513imtr4d 283 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → ((𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)))
5352impr 650 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ (𝐹:𝑋𝑌 ∧ (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
5414, 53impbida 913 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2140  wne 2933  wral 3051  wrex 3052  wss 3716  c0 4059  {csn 4322   cuni 4589  cima 5270  wf 6046  cfv 6050  (class class class)co 6815  Topctop 20921  TopOnctopon 20938  neicnei 21124   CnP ccnp 21252  Filcfil 21871   fLim cflim 21960   fLimf cflf 21961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336  df-map 8028  df-fbas 19966  df-fg 19967  df-top 20922  df-topon 20939  df-ntr 21047  df-nei 21125  df-cnp 21255  df-fil 21872  df-fm 21964  df-flim 21965  df-flf 21966
This theorem is referenced by:  cnpflf  22027
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