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Theorem cnpflf2 21556
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 20806 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
213expa 1256 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
323adantl3 1211 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹:𝑋𝑌)
4 simpl1 1056 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐽 ∈ (TopOn‘𝑋))
5 simpl3 1058 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴𝑋)
6 neiflim 21530 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴})))
7 cnpflf2.3 . . . . . . 7 𝐿 = ((nei‘𝐽)‘{𝐴})
87oveq2i 6538 . . . . . 6 (𝐽 fLim 𝐿) = (𝐽 fLim ((nei‘𝐽)‘{𝐴}))
96, 8syl6eleqr 2698 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ (𝐽 fLim 𝐿))
104, 5, 9syl2anc 690 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐴 ∈ (𝐽 fLim 𝐿))
11 simpr 475 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
12 cnpflfi 21555 . . . 4 ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
1310, 11, 12syl2anc 690 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))
143, 13jca 552 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹:𝑋𝑌 ∧ (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹)))
15 simpl1 1056 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐽 ∈ (TopOn‘𝑋))
16 topontop 20483 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1715, 16syl 17 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐽 ∈ Top)
18 simpl3 1058 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐴𝑋)
19 toponuni 20484 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2015, 19syl 17 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝑋 = 𝐽)
2118, 20eleqtrd 2689 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐴 𝐽)
227eleq2i 2679 . . . . . . . . . . . 12 (𝑧𝐿𝑧 ∈ ((nei‘𝐽)‘{𝐴}))
23 eqid 2609 . . . . . . . . . . . . 13 𝐽 = 𝐽
2423isneip 20661 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → (𝑧 ∈ ((nei‘𝐽)‘{𝐴}) ↔ (𝑧 𝐽 ∧ ∃𝑣𝐽 (𝐴𝑣𝑣𝑧))))
2522, 24syl5bb 270 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝐴 𝐽) → (𝑧𝐿 ↔ (𝑧 𝐽 ∧ ∃𝑣𝐽 (𝐴𝑣𝑣𝑧))))
2617, 21, 25syl2anc 690 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (𝑧𝐿 ↔ (𝑧 𝐽 ∧ ∃𝑣𝐽 (𝐴𝑣𝑣𝑧))))
27 imass2 5407 . . . . . . . . . . . . . . 15 (𝑣𝑧 → (𝐹𝑣) ⊆ (𝐹𝑧))
28 sstr2 3574 . . . . . . . . . . . . . . . 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 586 . . . . . . . . . . . . 13 ((𝐹𝑧) ⊆ 𝑢 → ((𝐴𝑣𝑣𝑧) → (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3231reximdv 2998 . . . . . . . . . . . 12 ((𝐹𝑧) ⊆ 𝑢 → (∃𝑣𝐽 (𝐴𝑣𝑣𝑧) → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3332com12 32 . . . . . . . . . . 11 (∃𝑣𝐽 (𝐴𝑣𝑣𝑧) → ((𝐹𝑧) ⊆ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3433adantl 480 . . . . . . . . . 10 ((𝑧 𝐽 ∧ ∃𝑣𝐽 (𝐴𝑣𝑣𝑧)) → ((𝐹𝑧) ⊆ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3526, 34syl6bi 241 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (𝑧𝐿 → ((𝐹𝑧) ⊆ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))))
3635rexlimdv 3011 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (∃𝑧𝐿 (𝐹𝑧) ⊆ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))
3736imim2d 54 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → (((𝐹𝐴) ∈ 𝑢 → ∃𝑧𝐿 (𝐹𝑧) ⊆ 𝑢) → ((𝐹𝐴) ∈ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢))))
3837ralimdv 2945 . . . . . 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 475 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐹:𝑋𝑌)
4038, 39jctild 563 . . . . 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 481 . . . 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 1057 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐾 ∈ (TopOn‘𝑌))
4318snssd 4280 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → {𝐴} ⊆ 𝑋)
44 snnzg 4250 . . . . . . . 8 (𝐴𝑋 → {𝐴} ≠ ∅)
4518, 44syl 17 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → {𝐴} ≠ ∅)
46 neifil 21436 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
4715, 43, 45, 46syl3anc 1317 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
487, 47syl5eqel 2691 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → 𝐿 ∈ (Fil‘𝑋))
49 isflf 21549 . . . . 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 1317 . . . 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 20793 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢𝐾 ((𝐹𝐴) ∈ 𝑢 → ∃𝑣𝐽 (𝐴𝑣 ∧ (𝐹𝑣) ⊆ 𝑢)))))
5251adantr 479 . . . 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 281 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ 𝐹:𝑋𝑌) → ((𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)))
5453impr 646 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ (𝐹:𝑋𝑌 ∧ (𝐹𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹))) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))
5514, 54impbida 872 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  wss 3539  c0 3873  {csn 4124   cuni 4366  cima 5031  wf 5786  cfv 5790  (class class class)co 6527  Topctop 20459  TopOnctopon 20460  neicnei 20653   CnP ccnp 20781  Filcfil 21401   fLim cflim 21490   fLimf cflf 21491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-map 7723  df-fbas 19510  df-fg 19511  df-top 20463  df-topon 20465  df-ntr 20576  df-nei 20654  df-cnp 20784  df-fil 21402  df-fm 21494  df-flim 21495  df-flf 21496
This theorem is referenced by:  cnpflf  21557
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