Theorem cnpval 14785

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 14782	. . . . 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 5601	. . . 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 2207	. . . 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 5599	. . . . . . . 8 ⊢ (𝑣 = 𝑃 → (𝑓‘𝑣) = (𝑓‘𝑃))
6	5	eleq1d 2276	. . . . . . 7 ⊢ (𝑣 = 𝑃 → ((𝑓‘𝑣) ∈ 𝑦 ↔ (𝑓‘𝑃) ∈ 𝑦))
7		eleq1 2270	. . . . . . . . 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 2509	. . . . . . 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 2508	. . . . 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 2765	. . . 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 6765	. . . . . 6 ⊢ ↑𝑚 Fn (V × V)
15		toponmax 14612	. . . . . . . 8 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ 𝐾)
16	15	elexd 2790	. . . . . . 7 ⊢ (𝐾 ∈ (TopOn‘𝑌) → 𝑌 ∈ V)
17	16	ad2antlr 489	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝑌 ∈ V)
18		toponmax 14612	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
19	18	elexd 2790	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ V)
20	19	ad2antrr 488	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃 ∈ 𝑋) → 𝑋 ∈ V)
21		fnovex 6000	. . . . . 6 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝑌 ∈ V ∧ 𝑋 ∈ V) → (𝑌 ↑𝑚 𝑋) ∈ V)
22	14, 17, 20, 21	mp3an2i 1355	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝑃 ∈ 𝑋) → (𝑌 ↑𝑚 𝑋) ∈ V)
23		rabexg 4203	. . . . 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 5696	. . 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 2240	. 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 1197	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 981   = wceq 1373   ∈ wcel 2178  ∀wral 2486  ∃wrex 2487  {crab 2490  Vcvv 2776   ⊆ wss 3174   ↦ cmpt 4121   × cxp 4691   “ cima 4696   Fn wfn 5285  ‘cfv 5290  (class class class)co 5967   ↑_𝑚 cmap 6758  TopOnctopon 14597   CnP ccnp 14773

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 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

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 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-map 6760  df-top 14585  df-topon 14598  df-cnp 14776

This theorem is referenced by:  iscnp  14786