Theorem cnpdis 23249

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 777	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → {𝐴} ∈ 𝐽)
2		simpll3 1216	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → 𝐴 ∈ 𝑋)
3		snidg 4619	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ {𝐴})
4	2, 3	syl 17	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → 𝐴 ∈ {𝐴})
5		simprr 773	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → (𝑓‘𝐴) ∈ 𝑥)
6		simplrr 778	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → 𝑓:𝑋⟶𝑌)
7		ffn 6670	. . . . . . . . . . 11 ⊢ (𝑓:𝑋⟶𝑌 → 𝑓 Fn 𝑋)
8		elpreima 7012	. . . . . . . . . . 11 ⊢ (𝑓 Fn 𝑋 → (𝐴 ∈ (◡𝑓 “ 𝑥) ↔ (𝐴 ∈ 𝑋 ∧ (𝑓‘𝐴) ∈ 𝑥)))
9	6, 7, 8	3syl 18	. . . . . . . . . 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 714	. . . . . . . . 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 4767	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → {𝐴} ⊆ (◡𝑓 “ 𝑥))
12		eleq2 2826	. . . . . . . . . 10 ⊢ (𝑦 = {𝐴} → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ {𝐴}))
13		sseq1 3961	. . . . . . . . . 10 ⊢ (𝑦 = {𝐴} → (𝑦 ⊆ (◡𝑓 “ 𝑥) ↔ {𝐴} ⊆ (◡𝑓 “ 𝑥)))
14	12, 13	anbi12d 633	. . . . . . . . 9 ⊢ (𝑦 = {𝐴} → ((𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ (◡𝑓 “ 𝑥)) ↔ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (◡𝑓 “ 𝑥))))
15	14	rspcev 3578	. . . . . . . 8 ⊢ (({𝐴} ∈ 𝐽 ∧ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (◡𝑓 “ 𝑥))) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ (◡𝑓 “ 𝑥)))
16	1, 4, 11, 15	syl12anc 837	. . . . . . 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 456	. . . . . 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 3130	. . . . 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 456	. . . 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 561	. . 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 1194	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
22		toponmax 22882	. . . . 5 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
23	21, 22	syl 17	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → 𝑌 ∈ 𝐾)
24		simpl1 1193	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25		toponmax 22882	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
26	24, 25	syl 17	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → 𝑋 ∈ 𝐽)
27	23, 26	elmapd 8789	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ (𝑌 ↑m 𝑋) ↔ 𝑓:𝑋⟶𝑌))
28		iscnp3 23200	. . . 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 480	. . 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 312	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ 𝑓 ∈ (𝑌 ↑m 𝑋)))
31	30	eqrdv 2735	1 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → ((𝐽 CnP 𝐾)‘𝐴) = (𝑌 ↑m 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ⊆ wss 3903  {csn 4582  ^◡ccnv 5631   “ cima 5635   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ↑_m cmap 8775  TopOnctopon 22866   CnP ccnp 23181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-map 8777  df-top 22850  df-topon 22867  df-cnp 23184

This theorem is referenced by: (None)