Theorem fcfval 24171

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 24080	. . . . 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 782	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → 𝑗 = 𝐽)
4	3	unieqd 4886	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ∪ 𝑗 = ∪ 𝐽)
5		toponuni 23052	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
6	5	ad2antrr 738	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → 𝑋 = ∪ 𝐽)
7	4, 6	eqtr4d 2801	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ∪ 𝑗 = 𝑋)
8		unieq 4884	. . . . . . . 8 ⊢ (𝑓 = 𝐿 → ∪ 𝑓 = ∪ 𝐿)
9	8	ad2antll 741	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ∪ 𝑓 = ∪ 𝐿)
10		filunibas 24019	. . . . . . . 8 ⊢ (𝐿 ∈ (Fil‘𝑌) → ∪ 𝐿 = 𝑌)
11	10	ad2antlr 739	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ∪ 𝐿 = 𝑌)
12	9, 11	eqtrd 2798	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ∪ 𝑓 = 𝑌)
13	7, 12	oveq12d 7430	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → (∪ 𝑗 ↑m ∪ 𝑓) = (𝑋 ↑m 𝑌))
14	7	oveq1d 7427	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → (∪ 𝑗 FilMap 𝑔) = (𝑋 FilMap 𝑔))
15		simprr 784	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → 𝑓 = 𝐿)
16	14, 15	fveq12d 6890	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ((∪ 𝑗 FilMap 𝑔)‘𝑓) = ((𝑋 FilMap 𝑔)‘𝐿))
17	3, 16	oveq12d 7430	. . . . 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 5199	. . . 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 23051	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
20	19	adantr 485	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐽 ∈ Top)
21		fvssunirn 6914	. . . . . 6 ⊢ (Fil‘𝑌) ⊆ ∪ ran Fil
22	21	sseli 3934	. . . . 5 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ ∪ ran Fil)
23	22	adantl 486	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐿 ∈ ∪ ran Fil)
24		ovex 7445	. . . . . 6 ⊢ (𝑋 ↑m 𝑌) ∈ V
25	24	mptex 7223	. . . . 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 7564	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fClusf 𝐿) = (𝑔 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿))))
28	27	3adant3 1150	. 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 489	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑔 = 𝐹) → 𝑔 = 𝐹)
30	29	oveq2d 7428	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑔 = 𝐹) → (𝑋 FilMap 𝑔) = (𝑋 FilMap 𝐹))
31	30	fveq1d 6885	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑔 = 𝐹) → ((𝑋 FilMap 𝑔)‘𝐿) = ((𝑋 FilMap 𝐹)‘𝐿))
32	31	oveq2d 7428	. 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 23064	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
34		filtop 23993	. . . 4 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝑌 ∈ 𝐿)
35		elmapg 8837	. . . 4 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐿) → (𝐹 ∈ (𝑋 ↑m 𝑌) ↔ 𝐹:𝑌⟶𝑋))
36	33, 34, 35	syl2an 607	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐹 ∈ (𝑋 ↑m 𝑌) ↔ 𝐹:𝑌⟶𝑋))
37	36	biimp3ar 1499	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐹 ∈ (𝑋 ↑m 𝑌))
38		ovexd 7447	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)) ∈ V)
39	28, 32, 37, 38	fvmptd 6999	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 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ∪ cuni 4873   ↦ cmpt 5193  ran crn 5664  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414   ↑_m cmap 8825  Topctop 23031  TopOnctopon 23048  Filcfil 23983   FilMap cfm 24071   fClus cfcls 24074   fClusf cfcf 24075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8827  df-fbas 21500  df-top 23032  df-topon 23049  df-fil 23984  df-fcf 24080

This theorem is referenced by:  isfcf  24172  fcfelbas  24174  flfssfcf  24176  uffcfflf  24177  cnpfcfi  24178  cnpfcf  24179