Theorem cnpval 23462

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 23460	. . . 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 6884	. . 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 6882	. . . . . . . 8 ⊢ (𝑣 = 𝑃 → (𝑓‘𝑣) = (𝑓‘𝑃))
4	3	eleq1d 2847	. . . . . . 7 ⊢ (𝑣 = 𝑃 → ((𝑓‘𝑣) ∈ 𝑦 ↔ (𝑓‘𝑃) ∈ 𝑦))
5		eleq1 2850	. . . . . . . . 9 ⊢ (𝑣 = 𝑃 → (𝑣 ∈ 𝑥 ↔ 𝑃 ∈ 𝑥))
6	5	anbi1d 643	. . . . . . . 8 ⊢ (𝑣 = 𝑃 → ((𝑣 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦) ↔ (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦)))
7	6	rexbidv 3188	. . . . . . 7 ⊢ (𝑣 = 𝑃 → (∃𝑥 ∈ 𝐽 (𝑣 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦)))
8	4, 7	imbi12d 347	. . . . . 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 3187	. . . . 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 3421	. . . 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 2762	. . . 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 7449	. . . . 5 ⊢ (𝑌 ↑m 𝑋) ∈ V
13	12	rabex 5307	. . . 4 ⊢ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))} ∈ V
14	10, 11, 13	fvmpt 6990	. . 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 2817	. 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 1127	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))})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  {crab 3414   ⊆ wss 3902   ↦ cmpt 5190   “ cima 5662  ‘cfv 6537  (class class class)co 7416   ↑_m cmap 8829  TopOnctopon 23136   CnP ccnp 23451

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-top 23120  df-topon 23137  df-cnp 23454

This theorem is used by:  iscnp  23463