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Theorem cnpdis 21585
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 773 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → {𝐴} ∈ 𝐽)
2 simpll3 1207 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝐴𝑋)
3 snidg 4504 . . . . . . . . 9 (𝐴𝑋𝐴 ∈ {𝐴})
42, 3syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝐴 ∈ {𝐴})
5 simprr 769 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → (𝑓𝐴) ∈ 𝑥)
6 simplrr 774 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝑓:𝑋𝑌)
7 ffn 6382 . . . . . . . . . . 11 (𝑓:𝑋𝑌𝑓 Fn 𝑋)
8 elpreima 6693 . . . . . . . . . . 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 709 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → 𝐴 ∈ (𝑓𝑥))
1110snssd 4649 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → {𝐴} ⊆ (𝑓𝑥))
12 eleq2 2871 . . . . . . . . . 10 (𝑦 = {𝐴} → (𝐴𝑦𝐴 ∈ {𝐴}))
13 sseq1 3913 . . . . . . . . . 10 (𝑦 = {𝐴} → (𝑦 ⊆ (𝑓𝑥) ↔ {𝐴} ⊆ (𝑓𝑥)))
1412, 13anbi12d 630 . . . . . . . . 9 (𝑦 = {𝐴} → ((𝐴𝑦𝑦 ⊆ (𝑓𝑥)) ↔ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (𝑓𝑥))))
1514rspcev 3559 . . . . . . . 8 (({𝐴} ∈ 𝐽 ∧ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (𝑓𝑥))) → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥)))
161, 4, 11, 15syl12anc 833 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ (𝑥𝐾 ∧ (𝑓𝐴) ∈ 𝑥)) → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥)))
1716expr 457 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) ∧ 𝑥𝐾) → ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))
1817ralrimiva 3149 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ ({𝐴} ∈ 𝐽𝑓:𝑋𝑌)) → ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))
1918expr 457 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓:𝑋𝑌 → ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥)))))
2019pm4.71d 562 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓:𝑋𝑌 ↔ (𝑓:𝑋𝑌 ∧ ∀𝑥𝐾 ((𝑓𝐴) ∈ 𝑥 → ∃𝑦𝐽 (𝐴𝑦𝑦 ⊆ (𝑓𝑥))))))
21 simpl2 1185 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
22 toponmax 21218 . . . . 5 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
2321, 22syl 17 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝑌𝐾)
24 simpl1 1184 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25 toponmax 21218 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
2624, 25syl 17 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → 𝑋𝐽)
2723, 26elmapd 8270 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ (𝑌𝑚 𝑋) ↔ 𝑓:𝑋𝑌))
28 iscnp3 21536 . . . 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 313 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ 𝑓 ∈ (𝑌𝑚 𝑋)))
3130eqrdv 2793 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 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wral 3105  wrex 3106  wss 3859  {csn 4472  ccnv 5442  cima 5446   Fn wfn 6220  wf 6221  cfv 6225  (class class class)co 7016  𝑚 cmap 8256  TopOnctopon 21202   CnP ccnp 21517
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-map 8258  df-top 21186  df-topon 21203  df-cnp 21520
This theorem is referenced by: (None)
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