Theorem cnpdis 13827

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 1038	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → 𝐴 ∈ 𝑋)
3		snidg 3623	. . . . . . . . 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 5367	. . . . . . . . . . 11 ⊢ (𝑓:𝑋⟶𝑌 → 𝑓 Fn 𝑋)
8		elpreima 5637	. . . . . . . . . . 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 944	. . . . . . . . 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 3739	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → {𝐴} ⊆ (◡𝑓 “ 𝑥))
12		eleq2 2241	. . . . . . . . . 10 ⊢ (𝑦 = {𝐴} → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ {𝐴}))
13		sseq1 3180	. . . . . . . . . 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 2843	. . . . . . . 8 ⊢ (({𝐴} ∈ 𝐽 ∧ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (◡𝑓 “ 𝑥))) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ (◡𝑓 “ 𝑥)))
16	1, 4, 11, 15	syl12anc 1236	. . . . . . 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 2550	. . . . 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 1001	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
22		toponmax 13610	. . . . 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 1000	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25		toponmax 13610	. . . . 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 6664	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ (𝑌 ↑𝑚 𝑋) ↔ 𝑓:𝑋⟶𝑌))
28		iscnp3 13788	. . . 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 2175	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 978   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456   ⊆ wss 3131  {csn 3594  ^◡ccnv 4627   “ cima 4631   Fn wfn 5213  ⟶wf 5214  ‘cfv 5218  (class class class)co 5877   ↑_𝑚 cmap 6650  TopOnctopon 13595   CnP ccnp 13771

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-map 6652  df-top 13583  df-topon 13596  df-cnp 13774

This theorem is referenced by: (None)