Theorem cnpval 23245

Description: The set of all functions from topology 𝐽 to topology 𝐾 that are continuous at a point 𝑃. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Assertion

Ref	Expression
cnpval	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))})

Distinct variable groups:   𝑥,𝑓,𝑦,𝐽   𝑓,𝐾,𝑥,𝑦   𝑓,𝑋,𝑥,𝑦   𝑃,𝑓,𝑥,𝑦   𝑓,𝑌,𝑥,𝑦

Proof of Theorem cnpval

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnpfval 23243	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 CnP 𝐾) = (𝑣 ∈ 𝑋 ↦ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑣) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑣 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))}))
2	1	fveq1d 6907	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐽 CnP 𝐾)‘𝑃) = ((𝑣 ∈ 𝑋 ↦ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑣) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑣 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))})‘𝑃))
3		fveq2 6905	. . . . . . . 8 ⊢ (𝑣 = 𝑃 → (𝑓‘𝑣) = (𝑓‘𝑃))
4	3	eleq1d 2825	. . . . . . 7 ⊢ (𝑣 = 𝑃 → ((𝑓‘𝑣) ∈ 𝑦 ↔ (𝑓‘𝑃) ∈ 𝑦))
5		eleq1 2828	. . . . . . . . 9 ⊢ (𝑣 = 𝑃 → (𝑣 ∈ 𝑥 ↔ 𝑃 ∈ 𝑥))
6	5	anbi1d 631	. . . . . . . 8 ⊢ (𝑣 = 𝑃 → ((𝑣 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦) ↔ (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦)))
7	6	rexbidv 3178	. . . . . . 7 ⊢ (𝑣 = 𝑃 → (∃𝑥 ∈ 𝐽 (𝑣 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦)))
8	4, 7	imbi12d 344	. . . . . 6 ⊢ (𝑣 = 𝑃 → (((𝑓‘𝑣) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑣 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦)) ↔ ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))))
9	8	ralbidv 3177	. . . . 5 ⊢ (𝑣 = 𝑃 → (∀𝑦 ∈ 𝐾 ((𝑓‘𝑣) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑣 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦)) ↔ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))))
10	9	rabbidv 3443	. . . 4 ⊢ (𝑣 = 𝑃 → {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑣) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑣 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))} = {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))})
11		eqid 2736	. . . 4 ⊢ (𝑣 ∈ 𝑋 ↦ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑣) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑣 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))}) = (𝑣 ∈ 𝑋 ↦ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑣) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑣 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))})
12		ovex 7465	. . . . 5 ⊢ (𝑌 ↑m 𝑋) ∈ V
13	12	rabex 5338	. . . 4 ⊢ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))} ∈ V
14	10, 11, 13	fvmpt 7015	. . 3 ⊢ (𝑃 ∈ 𝑋 → ((𝑣 ∈ 𝑋 ↦ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑣) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑣 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))})‘𝑃) = {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))})
15	2, 14	sylan9eq 2796	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃 ∈ 𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))})
16	15	3impa 1109	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃 ∈ 𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3060  ∃wrex 3069  {crab 3435   ⊆ wss 3950   ↦ cmpt 5224   “ cima 5687  ‘cfv 6560  (class class class)co 7432   ↑_m cmap 8867  TopOnctopon 22917   CnP ccnp 23234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-top 22901  df-topon 22918  df-cnp 23237

This theorem is referenced by:  iscnp  23246