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Theorem cnpdis 23190
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 776 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → {𝐴} ∈ 𝐽)
2 simpll3 1212 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝐴𝑋)
3 snidg 4658 . . . . . . . . 9 (𝐴𝑋𝐴 ∈ {𝐴})
42, 3syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝐴 ∈ {𝐴})
5 simprr 772 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → (𝑓𝐴) ∈ 𝑥)
6 simplrr 777 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝑓:𝑋𝑌)
7 ffn 6716 . . . . . . . . . . 11 (𝑓:𝑋𝑌𝑓 Fn 𝑋)
8 elpreima 7061 . . . . . . . . . . 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 712 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝐴 ∈ (𝑓𝑥))
1110snssd 4808 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → {𝐴} ⊆ (𝑓𝑥))
12 eleq2 2817 . . . . . . . . . 10 (𝑦 = {𝐴} → (𝐴𝑦𝐴 ∈ {𝐴}))
13 sseq1 4003 . . . . . . . . . 10 (𝑦 = {𝐴} → (𝑦 ⊆ (𝑓𝑥) ↔ {𝐴} ⊆ (𝑓𝑥)))
1412, 13anbi12d 630 . . . . . . . . 9 (𝑦 = {𝐴} → ((𝐴𝑦𝑦 ⊆ (𝑓𝑥)) ↔ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (𝑓𝑥))))
1514rspcev 3607 . . . . . . . 8 (({𝐴} ∈ 𝐽 ∧ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (𝑓𝑥))) → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥)))
161, 4, 11, 15syl12anc 836 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥)))
1716expr 456 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ 𝑥𝐾) → ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))
1817ralrimiva 3141 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) → ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))
1918expr 456 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓:𝑋𝑌 → ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥)))))
2019pm4.71d 561 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓:𝑋𝑌 ↔ (𝑓:𝑋𝑌 ∧ ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))))
21 simpl2 1190 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
22 toponmax 22821 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
2321, 22syl 17 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝑌𝐾)
24 simpl1 1189 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25 toponmax 22821 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
2624, 25syl 17 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝑋𝐽)
2723, 26elmapd 8852 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ (𝑌m 𝑋) ↔ 𝑓:𝑋𝑌))
28 iscnp3 23141 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) → (𝑓 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝑓:𝑋𝑌 ∧ ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))))
2928adantr 480 . . 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 312 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ 𝑓 ∈ (𝑌m 𝑋)))
3130eqrdv 2725 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 205  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3056  wrex 3065  wss 3944  {csn 4624  ccnv 5671  cima 5675   Fn wfn 6537  wf 6538  cfv 6542  (class class class)co 7414  m cmap 8838  TopOnctopon 22805   CnP ccnp 23122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8840  df-top 22789  df-topon 22806  df-cnp 23125
This theorem is referenced by: (None)
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