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Theorem cnpdis 20907
 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 796 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → {𝐴} ∈ 𝐽)
2 simpll3 1095 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝐴𝑋)
3 snidg 4153 . . . . . . . . 9 (𝐴𝑋𝐴 ∈ {𝐴})
42, 3syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝐴 ∈ {𝐴})
5 simprr 792 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → (𝑓𝐴) ∈ 𝑥)
6 simplrr 797 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝑓:𝑋𝑌)
7 ffn 5958 . . . . . . . . . . 11 (𝑓:𝑋𝑌𝑓 Fn 𝑋)
8 elpreima 6245 . . . . . . . . . . 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 959 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝐴 ∈ (𝑓𝑥))
1110snssd 4281 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → {𝐴} ⊆ (𝑓𝑥))
12 eleq2 2677 . . . . . . . . . 10 (𝑦 = {𝐴} → (𝐴𝑦𝐴 ∈ {𝐴}))
13 sseq1 3589 . . . . . . . . . 10 (𝑦 = {𝐴} → (𝑦 ⊆ (𝑓𝑥) ↔ {𝐴} ⊆ (𝑓𝑥)))
1412, 13anbi12d 743 . . . . . . . . 9 (𝑦 = {𝐴} → ((𝐴𝑦𝑦 ⊆ (𝑓𝑥)) ↔ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (𝑓𝑥))))
1514rspcev 3282 . . . . . . . 8 (({𝐴} ∈ 𝐽 ∧ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (𝑓𝑥))) → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥)))
161, 4, 11, 15syl12anc 1316 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥)))
1716expr 641 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ 𝑥𝐾) → ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))
1817ralrimiva 2949 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) → ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))
1918expr 641 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓:𝑋𝑌 → ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥)))))
2019pm4.71d 664 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓:𝑋𝑌 ↔ (𝑓:𝑋𝑌 ∧ ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))))
21 simpl2 1058 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
22 toponmax 20543 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
2321, 22syl 17 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝑌𝐾)
24 simpl1 1057 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25 toponmax 20543 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
2624, 25syl 17 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝑋𝐽)
2723, 26elmapd 7758 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ (𝑌𝑚 𝑋) ↔ 𝑓:𝑋𝑌))
28 iscnp3 20858 . . . 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 300 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ 𝑓 ∈ (𝑌𝑚 𝑋)))
3130eqrdv 2608 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  {csn 4125  ◡ccnv 5037   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  TopOnctopon 20518   CnP ccnp 20839 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-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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  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-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-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-top 20521  df-topon 20523  df-cnp 20842 This theorem is referenced by: (None)
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