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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 (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 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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