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Theorem cnpdis 21002
Description: If 𝐴 is an isolated point in 𝑋 (or equivalently, the singleton {𝐴} is open in 𝑋), then every function is continuous at 𝐴. (Contributed by Mario Carneiro, 9-Sep-2015.)
Assertion
Ref Expression
cnpdis (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → ((𝐽 CnP 𝐾)‘𝐴) = (𝑌𝑚 𝑋))

Proof of Theorem cnpdis
Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplrl 799 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → {𝐴} ∈ 𝐽)
2 simpll3 1100 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝐴𝑋)
3 snidg 4182 . . . . . . . . 9 (𝐴𝑋𝐴 ∈ {𝐴})
42, 3syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝐴 ∈ {𝐴})
5 simprr 795 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → (𝑓𝐴) ∈ 𝑥)
6 simplrr 800 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝑓:𝑋𝑌)
7 ffn 6004 . . . . . . . . . . 11 (𝑓:𝑋𝑌𝑓 Fn 𝑋)
8 elpreima 6294 . . . . . . . . . . 11 (𝑓 Fn 𝑋 → (𝐴 ∈ (𝑓𝑥) ↔ (𝐴𝑋 ∧ (𝑓𝐴) ∈ 𝑥)))
96, 7, 83syl 18 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → (𝐴 ∈ (𝑓𝑥) ↔ (𝐴𝑋 ∧ (𝑓𝐴) ∈ 𝑥)))
102, 5, 9mpbir2and 956 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝐴 ∈ (𝑓𝑥))
1110snssd 4314 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → {𝐴} ⊆ (𝑓𝑥))
12 eleq2 2693 . . . . . . . . . 10 (𝑦 = {𝐴} → (𝐴𝑦𝐴 ∈ {𝐴}))
13 sseq1 3610 . . . . . . . . . 10 (𝑦 = {𝐴} → (𝑦 ⊆ (𝑓𝑥) ↔ {𝐴} ⊆ (𝑓𝑥)))
1412, 13anbi12d 746 . . . . . . . . 9 (𝑦 = {𝐴} → ((𝐴𝑦𝑦 ⊆ (𝑓𝑥)) ↔ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (𝑓𝑥))))
1514rspcev 3300 . . . . . . . 8 (({𝐴} ∈ 𝐽 ∧ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (𝑓𝑥))) → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥)))
161, 4, 11, 15syl12anc 1321 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥)))
1716expr 642 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ 𝑥𝐾) → ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))
1817ralrimiva 2965 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) → ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))
1918expr 642 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓:𝑋𝑌 → ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥)))))
2019pm4.71d 665 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓:𝑋𝑌 ↔ (𝑓:𝑋𝑌 ∧ ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))))
21 simpl2 1063 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
22 toponmax 20638 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
2321, 22syl 17 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝑌𝐾)
24 simpl1 1062 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25 toponmax 20638 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
2624, 25syl 17 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝑋𝐽)
2723, 26elmapd 7817 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ (𝑌𝑚 𝑋) ↔ 𝑓:𝑋𝑌))
28 iscnp3 20953 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) → (𝑓 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝑓:𝑋𝑌 ∧ ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))))
2928adantr 481 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝑓:𝑋𝑌 ∧ ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))))
3020, 27, 293bitr4rd 301 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ 𝑓 ∈ (𝑌𝑚 𝑋)))
3130eqrdv 2624 1 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → ((𝐽 CnP 𝐾)‘𝐴) = (𝑌𝑚 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  wrex 2913  wss 3560  {csn 4153  ccnv 5078  cima 5082   Fn wfn 5845  wf 5846  cfv 5850  (class class class)co 6605  𝑚 cmap 7803  TopOnctopon 20613   CnP ccnp 20934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-top 20616  df-topon 20618  df-cnp 20937
This theorem is referenced by: (None)
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