Theorem fcfval 24041

Description: The set of cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Assertion

Ref	Expression
fcfval	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))

Proof of Theorem fcfval

Dummy variables 𝑓 𝑔 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fcf 23950	. . . . 5 ⊢ fClusf = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ (𝑔 ∈ (∪ 𝑗 ↑m ∪ 𝑓) ↦ (𝑗 fClus ((∪ 𝑗 FilMap 𝑔)‘𝑓))))
2	1	a1i 11	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → fClusf = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ (𝑔 ∈ (∪ 𝑗 ↑m ∪ 𝑓) ↦ (𝑗 fClus ((∪ 𝑗 FilMap 𝑔)‘𝑓)))))
3		simprl 771	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → 𝑗 = 𝐽)
4	3	unieqd 4920	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ∪ 𝑗 = ∪ 𝐽)
5		toponuni 22920	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
6	5	ad2antrr 726	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → 𝑋 = ∪ 𝐽)
7	4, 6	eqtr4d 2780	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ∪ 𝑗 = 𝑋)
8		unieq 4918	. . . . . . . 8 ⊢ (𝑓 = 𝐿 → ∪ 𝑓 = ∪ 𝐿)
9	8	ad2antll 729	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ∪ 𝑓 = ∪ 𝐿)
10		filunibas 23889	. . . . . . . 8 ⊢ (𝐿 ∈ (Fil‘𝑌) → ∪ 𝐿 = 𝑌)
11	10	ad2antlr 727	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ∪ 𝐿 = 𝑌)
12	9, 11	eqtrd 2777	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ∪ 𝑓 = 𝑌)
13	7, 12	oveq12d 7449	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → (∪ 𝑗 ↑m ∪ 𝑓) = (𝑋 ↑m 𝑌))
14	7	oveq1d 7446	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → (∪ 𝑗 FilMap 𝑔) = (𝑋 FilMap 𝑔))
15		simprr 773	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → 𝑓 = 𝐿)
16	14, 15	fveq12d 6913	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ((∪ 𝑗 FilMap 𝑔)‘𝑓) = ((𝑋 FilMap 𝑔)‘𝐿))
17	3, 16	oveq12d 7449	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → (𝑗 fClus ((∪ 𝑗 FilMap 𝑔)‘𝑓)) = (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿)))
18	13, 17	mpteq12dv 5233	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → (𝑔 ∈ (∪ 𝑗 ↑m ∪ 𝑓) ↦ (𝑗 fClus ((∪ 𝑗 FilMap 𝑔)‘𝑓))) = (𝑔 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿))))
19		topontop 22919	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
20	19	adantr 480	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐽 ∈ Top)
21		fvssunirn 6939	. . . . . 6 ⊢ (Fil‘𝑌) ⊆ ∪ ran Fil
22	21	sseli 3979	. . . . 5 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ ∪ ran Fil)
23	22	adantl 481	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐿 ∈ ∪ ran Fil)
24		ovex 7464	. . . . . 6 ⊢ (𝑋 ↑m 𝑌) ∈ V
25	24	mptex 7243	. . . . 5 ⊢ (𝑔 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿))) ∈ V
26	25	a1i 11	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝑔 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿))) ∈ V)
27	2, 18, 20, 23, 26	ovmpod 7585	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fClusf 𝐿) = (𝑔 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿))))
28	27	3adant3 1133	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐽 fClusf 𝐿) = (𝑔 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿))))
29		simpr 484	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑔 = 𝐹) → 𝑔 = 𝐹)
30	29	oveq2d 7447	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑔 = 𝐹) → (𝑋 FilMap 𝑔) = (𝑋 FilMap 𝐹))
31	30	fveq1d 6908	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑔 = 𝐹) → ((𝑋 FilMap 𝑔)‘𝐿) = ((𝑋 FilMap 𝐹)‘𝐿))
32	31	oveq2d 7447	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑔 = 𝐹) → (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿)) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))
33		toponmax 22932	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
34		filtop 23863	. . . 4 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝑌 ∈ 𝐿)
35		elmapg 8879	. . . 4 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐿) → (𝐹 ∈ (𝑋 ↑m 𝑌) ↔ 𝐹:𝑌⟶𝑋))
36	33, 34, 35	syl2an 596	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐹 ∈ (𝑋 ↑m 𝑌) ↔ 𝐹:𝑌⟶𝑋))
37	36	biimp3ar 1472	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐹 ∈ (𝑋 ↑m 𝑌))
38		ovexd 7466	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)) ∈ V)
39	28, 32, 37, 38	fvmptd 7023	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  Vcvv 3480  ∪ cuni 4907   ↦ cmpt 5225  ran crn 5686  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433   ↑_m cmap 8866  Topctop 22899  TopOnctopon 22916  Filcfil 23853   FilMap cfm 23941   fClus cfcls 23944   fClusf cfcf 23945

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-fbas 21361  df-top 22900  df-topon 22917  df-fil 23854  df-fcf 23950

This theorem is referenced by:  isfcf  24042  fcfelbas  24044  flfssfcf  24046  uffcfflf  24047  cnpfcfi  24048  cnpfcf  24049