Theorem cnpval 14921

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 14918	. . . . 5 ⊢ ((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 5641	. . . 4 ⊢ ((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	2	adantr 276	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃 ∈ 𝑋) → ((𝐽 CnP 𝐾)‘𝑃) = ((𝑣 ∈ 𝑋 ↦ {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑣) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑣 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))})‘𝑃))
4		eqid 2231	. . . 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))})
5		fveq2 5639	. . . . . . . 8 ⊢ (𝑣 = 𝑃 → (𝑓‘𝑣) = (𝑓‘𝑃))
6	5	eleq1d 2300	. . . . . . 7 ⊢ (𝑣 = 𝑃 → ((𝑓‘𝑣) ∈ 𝑦 ↔ (𝑓‘𝑃) ∈ 𝑦))
7		eleq1 2294	. . . . . . . . 9 ⊢ (𝑣 = 𝑃 → (𝑣 ∈ 𝑥 ↔ 𝑃 ∈ 𝑥))
8	7	anbi1d 465	. . . . . . . 8 ⊢ (𝑣 = 𝑃 → ((𝑣 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦) ↔ (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦)))
9	8	rexbidv 2533	. . . . . . 7 ⊢ (𝑣 = 𝑃 → (∃𝑥 ∈ 𝐽 (𝑣 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦) ↔ ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦)))
10	6, 9	imbi12d 234	. . . . . 6 ⊢ (𝑣 = 𝑃 → (((𝑓‘𝑣) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑣 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦)) ↔ ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))))
11	10	ralbidv 2532	. . . . 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))))
12	11	rabbidv 2791	. . . 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))})
13		simpr 110	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝑃 ∈ 𝑋)
14		fnmap 6823	. . . . . 6 ⊢ ↑𝑚 Fn (V × V)
15		toponmax 14748	. . . . . . . 8 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
16	15	elexd 2816	. . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ V)
17	16	ad2antlr 489	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝑌 ∈ V)
18		toponmax 14748	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
19	18	elexd 2816	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ V)
20	19	ad2antrr 488	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝑋 ∈ V)
21		fnovex 6050	. . . . . 6 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝑌 ∈ V ∧ 𝑋 ∈ V) → (𝑌 ↑𝑚 𝑋) ∈ V)
22	14, 17, 20, 21	mp3an2i 1378	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝑌 ↑𝑚 𝑋) ∈ V)
23		rabexg 4233	. . . . 5 ⊢ ((𝑌 ↑𝑚 𝑋) ∈ V → {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))} ∈ V)
24	22, 23	syl 14	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃 ∈ 𝑋) → {𝑓 ∈ (𝑌 ↑𝑚 𝑋) ∣ ∀𝑦 ∈ 𝐾 ((𝑓‘𝑃) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝑃 ∈ 𝑥 ∧ (𝑓 “ 𝑥) ⊆ 𝑦))} ∈ V)
25	4, 12, 13, 24	fvmptd3 5740	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 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))})
26	3, 25	eqtrd 2264	. 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))})
27	26	3impa 1220	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 104   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  {crab 2514  Vcvv 2802   ⊆ wss 3200   ↦ cmpt 4150   × cxp 4723   “ cima 4728   Fn wfn 5321  ‘cfv 5326  (class class class)co 6017   ↑_𝑚 cmap 6816  TopOnctopon 14733   CnP ccnp 14909

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-map 6818  df-top 14721  df-topon 14734  df-cnp 14912

This theorem is referenced by:  iscnp  14922