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Theorem cnpflf2 21614
 Description: 𝐹 is continuous at point 𝐴 iff a limit of 𝐹 when 𝑥 tends to 𝐴 is (𝐹‘𝐴). 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 20864 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
213expa 1257 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
323adantl3 1212 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
4 simpl1 1057 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
5 simpl3 1059 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴𝑋)
6 neiflim 21588 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))
7 cnpflf2.3 . . . . . . 7 𝐿 = ((nei‘𝐽)‘{𝐴})
87oveq2i 6560 . . . . . 6 (𝐽 fLim 𝐿) = (𝐽 fLim ((nei‘𝐽)‘{𝐴}))
96, 8syl6eleqr 2699 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ (𝐽 fLim 𝐿))
104, 5, 9syl2anc 691 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fLim 𝐿))
11 simpr 476 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
12 cnpflfi 21613 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
1310, 11, 12syl2anc 691 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
143, 13jca 553 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹:𝑋𝑌 ∧ (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹)))
15 simpl1 1057 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐽 ∈ (TopOn‘𝑋))
16 topontop 20541 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1715, 16syl 17 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐽 ∈ Top)
18 simpl3 1059 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐴𝑋)
19 toponuni 20542 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2015, 19syl 17 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝑋 = 𝐽)
2118, 20eleqtrd 2690 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐴 𝐽)
227eleq2i 2680 . . . . . . . . . . . 12 (𝑧𝐿𝑧 ∈ ((nei‘𝐽)‘{𝐴}))
23 eqid 2610 . . . . . . . . . . . . 13 𝐽 = 𝐽
2423isneip 20719 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → (𝑧 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑧 𝐽 ∧ ∃𝑣𝐽 (𝐴𝑣𝑣𝑧))))
2522, 24syl5bb 271 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → (𝑧𝐿 ↔ (𝑧 𝐽 ∧ ∃𝑣𝐽 (𝐴𝑣𝑣𝑧))))
2617, 21, 25syl2anc 691 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (𝑧𝐿 ↔ (𝑧 𝐽 ∧ ∃𝑣𝐽 (𝐴𝑣𝑣𝑧))))
27 imass2 5420 . . . . . . . . . . . . . . 15 (𝑣𝑧 → (𝐹𝑣) ⊆ (𝐹𝑧))
28 sstr2 3575 . . . . . . . . . . . . . . . 16 ((𝐹𝑣) ⊆ (𝐹𝑧) → ((𝐹𝑧) ⊆ 𝑢 → (𝐹𝑣) ⊆ 𝑢))
2928com12 32 . . . . . . . . . . . . . . 15 ((𝐹𝑧) ⊆ 𝑢 → ((𝐹𝑣) ⊆ (𝐹𝑧) → (𝐹𝑣) ⊆ 𝑢))
3027, 29syl5 33 . . . . . . . . . . . . . 14 ((𝐹𝑧) ⊆ 𝑢 → (𝑣𝑧 → (𝐹𝑣) ⊆ 𝑢))
3130anim2d 587 . . . . . . . . . . . . 13 ((𝐹𝑧) ⊆ 𝑢 → ((𝐴𝑣𝑣𝑧) → (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3231reximdv 2999 . . . . . . . . . . . 12 ((𝐹𝑧) ⊆ 𝑢 → (∃𝑣𝐽 (𝐴𝑣𝑣𝑧) → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3332com12 32 . . . . . . . . . . 11 (∃𝑣𝐽 (𝐴𝑣𝑣𝑧) → ((𝐹𝑧) ⊆ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3433adantl 481 . . . . . . . . . 10 ((𝑧 𝐽 ∧ ∃𝑣𝐽 (𝐴𝑣𝑣𝑧)) → ((𝐹𝑧) ⊆ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3526, 34syl6bi 242 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (𝑧𝐿 → ((𝐹𝑧) ⊆ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))))
3635rexlimdv 3012 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (∃𝑧𝐿 (𝐹𝑧) ⊆ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3736imim2d 55 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (((𝐹𝐴) ∈ 𝑢 → ∃𝑧𝐿 (𝐹𝑧) ⊆ 𝑢) → ((𝐹𝐴) ∈ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))))
3837ralimdv 2946 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑧𝐿 (𝐹𝑧) ⊆ 𝑢) → ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))))
39 simpr 476 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐹:𝑋𝑌)
4038, 39jctild 564 . . . . 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)))))
4140adantld 482 . . . 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)))))
42 simpl2 1058 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐾 ∈ (TopOn‘𝑌))
4318snssd 4281 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → {𝐴} ⊆ 𝑋)
44 snnzg 4251 . . . . . . . 8 (𝐴𝑋 → {𝐴} ≠ ∅)
4518, 44syl 17 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → {𝐴} ≠ ∅)
46 neifil 21494 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
4715, 43, 45, 46syl3anc 1318 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
487, 47syl5eqel 2692 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐿 ∈ (Fil‘𝑋))
49 isflf 21607 . . . . 5 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑋𝑌) → ((𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) ↔ ((𝐹𝐴) ∈ 𝑌 ∧ ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑧𝐿 (𝐹𝑧) ⊆ 𝑢))))
5042, 48, 39, 49syl3anc 1318 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → ((𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) ↔ ((𝐹𝐴) ∈ 𝑌 ∧ ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑧𝐿 (𝐹𝑧) ⊆ 𝑢))))
51 iscnp 20851 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))))
5251adantr 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)))))
5341, 50, 523imtr4d 282 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → ((𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)))
5453impr 647 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ (𝐹:𝑋𝑌 ∧ (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
5514, 54impbida 873 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ cuni 4372   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Topctop 20517  TopOnctopon 20518  neicnei 20711   CnP ccnp 20839  Filcfil 21459   fLim cflim 21548   fLimf cflf 21549 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-fbas 19564  df-fg 19565  df-top 20521  df-topon 20523  df-ntr 20634  df-nei 20712  df-cnp 20842  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554 This theorem is referenced by:  cnpflf  21615
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