Theorem cnpdis 14789

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 1041	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → 𝐴 ∈ 𝑋)
3		snidg 3667	. . . . . . . . 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 5435	. . . . . . . . . . 11 ⊢ (𝑓:𝑋⟶𝑌 → 𝑓 Fn 𝑋)
8		elpreima 5712	. . . . . . . . . . 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 947	. . . . . . . . 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 3784	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ ({𝐴} ∈ 𝐽 ∧ 𝑓:𝑋⟶𝑌)) ∧ (𝑥 ∈ 𝐾 ∧ (𝑓‘𝐴) ∈ 𝑥)) → {𝐴} ⊆ (◡𝑓 “ 𝑥))
12		eleq2 2270	. . . . . . . . . 10 ⊢ (𝑦 = {𝐴} → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ {𝐴}))
13		sseq1 3220	. . . . . . . . . 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 2881	. . . . . . . 8 ⊢ (({𝐴} ∈ 𝐽 ∧ (𝐴 ∈ {𝐴} ∧ {𝐴} ⊆ (◡𝑓 “ 𝑥))) → ∃𝑦 ∈ 𝐽 (𝐴 ∈ 𝑦 ∧ 𝑦 ⊆ (◡𝑓 “ 𝑥)))
16	1, 4, 11, 15	syl12anc 1248	. . . . . . 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 2580	. . . . 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 1004	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑌))
22		toponmax 14572	. . . . 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 1003	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → 𝐽 ∈ (TopOn‘𝑋))
25		toponmax 14572	. . . . 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 6762	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) ∧ {𝐴} ∈ 𝐽) → (𝑓 ∈ (𝑌 ↑𝑚 𝑋) ↔ 𝑓:𝑋⟶𝑌))
28		iscnp3 14750	. . . 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 2204	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 981   = wceq 1373   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486   ⊆ wss 3170  {csn 3638  ^◡ccnv 4682   “ cima 4686   Fn wfn 5275  ⟶wf 5276  ‘cfv 5280  (class class class)co 5957   ↑_𝑚 cmap 6748  TopOnctopon 14557   CnP ccnp 14733

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-map 6750  df-top 14545  df-topon 14558  df-cnp 14736

This theorem is referenced by: (None)