Theorem cnpval 23123

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 23121	. . . 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 6860	. . 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 6858	. . . . . . . 8 ⊢ (𝑣 = 𝑃 → (𝑓‘𝑣) = (𝑓‘𝑃))
4	3	eleq1d 2813	. . . . . . 7 ⊢ (𝑣 = 𝑃 → ((𝑓‘𝑣) ∈ 𝑦 ↔ (𝑓‘𝑃) ∈ 𝑦))
5		eleq1 2816	. . . . . . . . 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 3157	. . . . . . 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 3156	. . . . 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 3413	. . . 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 2729	. . . 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 7420	. . . . 5 ⊢ (𝑌 ↑m 𝑋) ∈ V
13	12	rabex 5294	. . . 4 ⊢ {𝑓 ∈ (𝑌 ↑m 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))} ∈ V
14	10, 11, 13	fvmpt 6968	. . 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 2784	. 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 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3405   ⊆ wss 3914   ↦ cmpt 5188   “ cima 5641  ‘cfv 6511  (class class class)co 7387   ↑_m cmap 8799  TopOnctopon 22797   CnP ccnp 23112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-top 22781  df-topon 22798  df-cnp 23115

This theorem is referenced by:  iscnp  23124