Theorem cnpdis 14410

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) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → ((𝐽 CnP 𝐾)‘𝐴) = (𝑌 ↑𝑚 𝑋))

Proof of Theorem cnpdis

Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplrl 535	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → {𝐴} ∈ 𝐽)
2		simpll3 1040	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → 𝐴 ∈ 𝑋)
3		snidg 3647	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ {𝐴})
4	2, 3	syl 14	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → 𝐴 ∈ {𝐴})
5		simprr 531	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → (𝑓‘𝐴) ∈ 𝑥)
6		simplrr 536	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → 𝑓:𝑋⟶𝑌)
7		ffn 5403	. . . . . . . . . . 11 ⊢ (𝑓:𝑋⟶𝑌 → 𝑓 Fn 𝑋)
8		elpreima 5677	. . . . . . . . . . 11 ⊢ (𝑓 Fn 𝑋 → (𝐴 ∈ (◡𝑓 “ 𝑥) ↔ (𝐴 ∈ 𝑋 ∧ (𝑓‘𝐴) ∈ 𝑥)))
9	6, 7, 8	3syl 17	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → (𝐴 ∈ (◡𝑓 “ 𝑥) ↔ (𝐴 ∈ 𝑋 ∧ (𝑓‘𝐴) ∈ 𝑥)))
10	2, 5, 9	mpbir2and 946	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → 𝐴 ∈ (◡𝑓 “ 𝑥))
11	10	snssd 3763	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → {𝐴} ⊆ (◡𝑓 “ 𝑥))
12		eleq2 2257	. . . . . . . . . 10 ⊢ (𝑦 = {𝐴} → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ {𝐴}))
13		sseq1 3202	. . . . . . . . . 10 ⊢ (𝑦 = {𝐴} → (𝑦 ⊆ (◡𝑓 “ 𝑥) ↔ {𝐴} ⊆ (◡𝑓 “ 𝑥)))
14	12, 13	anbi12d 473	. . . . . . . . 9 ⊢ (𝑦 = {𝐴} → ((𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ (◡𝑓 “ 𝑥)) ↔ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (◡𝑓 “ 𝑥))))
15	14	rspcev 2864	. . . . . . . 8 ⊢ (({𝐴} ∈ 𝐽 ∧ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (◡𝑓 “ 𝑥))) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ (◡𝑓 “ 𝑥)))
16	1, 4, 11, 15	syl12anc 1247	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ (◡𝑓 “ 𝑥)))
17	16	expr 375	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ 𝑥 ∈ 𝐾) → ((𝑓‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ (◡𝑓 “ 𝑥))))
18	17	ralrimiva 2567	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) → ∀𝑥 ∈ 𝐾 ((𝑓‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ (◡𝑓 “ 𝑥))))
19	18	expr 375	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓:𝑋⟶𝑌 → ∀𝑥 ∈ 𝐾 ((𝑓‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ (◡𝑓 “ 𝑥)))))
20	19	pm4.71d 393	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓:𝑋⟶𝑌 ↔ (𝑓:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 ((𝑓‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ (◡𝑓 “ 𝑥))))))
21		simpl2 1003	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
22		toponmax 14193	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
23	21, 22	syl 14	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → 𝑌 ∈ 𝐾)
24		simpl1 1002	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25		toponmax 14193	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
26	24, 25	syl 14	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → 𝑋 ∈ 𝐽)
27	23, 26	elmapd 6716	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ (𝑌 ↑𝑚 𝑋) ↔ 𝑓:𝑋⟶𝑌))
28		iscnp3 14371	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝑓 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝑓:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 ((𝑓‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ (◡𝑓 “ 𝑥))))))
29	28	adantr 276	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝑓:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 ((𝑓‘𝐴) ∈ 𝑥 → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ (◡𝑓 “ 𝑥))))))
30	20, 27, 29	3bitr4rd 221	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ 𝑓 ∈ (𝑌 ↑𝑚 𝑋)))
31	30	eqrdv 2191	1 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → ((𝐽 CnP 𝐾)‘𝐴) = (𝑌 ↑𝑚 𝑋))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473   ⊆ wss 3153  {csn 3618  ^◡ccnv 4658   “ cima 4662   Fn wfn 5249  ⟶wf 5250  ‘cfv 5254  (class class class)co 5918   ↑_𝑚 cmap 6702  TopOnctopon 14178   CnP ccnp 14354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-top 14166  df-topon 14179  df-cnp 14357

This theorem is referenced by: (None)