Theorem cnpdis 14125

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 1039	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → 𝐴 ∈ 𝑋)
3		snidg 3635	. . . . . . . . 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 5379	. . . . . . . . . . 11 ⊢ (𝑓:𝑋⟶𝑌 → 𝑓 Fn 𝑋)
8		elpreima 5650	. . . . . . . . . . 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 945	. . . . . . . . 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 3751	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → {𝐴} ⊆ (◡𝑓 “ 𝑥))
12		eleq2 2252	. . . . . . . . . 10 ⊢ (𝑦 = {𝐴} → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ {𝐴}))
13		sseq1 3192	. . . . . . . . . 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 2855	. . . . . . . 8 ⊢ (({𝐴} ∈ 𝐽 ∧ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (◡𝑓 “ 𝑥))) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ (◡𝑓 “ 𝑥)))
16	1, 4, 11, 15	syl12anc 1246	. . . . . . 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 2562	. . . . 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 1002	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
22		toponmax 13908	. . . . 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 1001	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25		toponmax 13908	. . . . 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 6679	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ (𝑌 ↑𝑚 𝑋) ↔ 𝑓:𝑋⟶𝑌))
28		iscnp3 14086	. . . 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 2186	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 979   = wceq 1363   ∈ wcel 2159  ∀wral 2467  ∃wrex 2468   ⊆ wss 3143  {csn 3606  ^◡ccnv 4639   “ cima 4643   Fn wfn 5225  ⟶wf 5226  ‘cfv 5230  (class class class)co 5890   ↑_𝑚 cmap 6665  TopOnctopon 13893   CnP ccnp 14069

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-ral 2472  df-rex 2473  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-fv 5238  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-map 6667  df-top 13881  df-topon 13894  df-cnp 14072

This theorem is referenced by: (None)