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Theorem cnpdis 21318
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 762 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → {𝐴} ∈ 𝐽)
2 simpll3 1258 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝐴𝑋)
3 snidg 4345 . . . . . . . . 9 (𝐴𝑋𝐴 ∈ {𝐴})
42, 3syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝐴 ∈ {𝐴})
5 simprr 756 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → (𝑓𝐴) ∈ 𝑥)
6 simplrr 763 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝑓:𝑋𝑌)
7 ffn 6185 . . . . . . . . . . 11 (𝑓:𝑋𝑌𝑓 Fn 𝑋)
8 elpreima 6480 . . . . . . . . . . 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 692 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝐴 ∈ (𝑓𝑥))
1110snssd 4475 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → {𝐴} ⊆ (𝑓𝑥))
12 eleq2 2839 . . . . . . . . . 10 (𝑦 = {𝐴} → (𝐴𝑦𝐴 ∈ {𝐴}))
13 sseq1 3775 . . . . . . . . . 10 (𝑦 = {𝐴} → (𝑦 ⊆ (𝑓𝑥) ↔ {𝐴} ⊆ (𝑓𝑥)))
1412, 13anbi12d 616 . . . . . . . . 9 (𝑦 = {𝐴} → ((𝐴𝑦𝑦 ⊆ (𝑓𝑥)) ↔ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (𝑓𝑥))))
1514rspcev 3460 . . . . . . . 8 (({𝐴} ∈ 𝐽 ∧ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (𝑓𝑥))) → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥)))
161, 4, 11, 15syl12anc 1474 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥)))
1716expr 444 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ 𝑥𝐾) → ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))
1817ralrimiva 3115 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) → ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))
1918expr 444 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓:𝑋𝑌 → ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥)))))
2019pm4.71d 551 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓:𝑋𝑌 ↔ (𝑓:𝑋𝑌 ∧ ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))))
21 simpl2 1229 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
22 toponmax 20951 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
2321, 22syl 17 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝑌𝐾)
24 simpl1 1227 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25 toponmax 20951 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
2624, 25syl 17 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝑋𝐽)
2723, 26elmapd 8023 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ (𝑌𝑚 𝑋) ↔ 𝑓:𝑋𝑌))
28 iscnp3 21269 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) → (𝑓 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝑓:𝑋𝑌 ∧ ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))))
2928adantr 466 . . 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 2769 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 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wrex 3062  wss 3723  {csn 4316  ccnv 5248  cima 5252   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  𝑚 cmap 8009  TopOnctopon 20935   CnP ccnp 21250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-top 20919  df-topon 20936  df-cnp 21253
This theorem is referenced by: (None)
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