Theorem fcfval 23282

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 23191	. . . . 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 768	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → 𝑗 = 𝐽)
4	3	unieqd 4865	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ∪ 𝑗 = ∪ 𝐽)
5		toponuni 22161	. . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
6	5	ad2antrr 723	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → 𝑋 = ∪ 𝐽)
7	4, 6	eqtr4d 2779	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ∪ 𝑗 = 𝑋)
8		unieq 4862	. . . . . . . 8 ⊢ (𝑓 = 𝐿 → ∪ 𝑓 = ∪ 𝐿)
9	8	ad2antll 726	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ∪ 𝑓 = ∪ 𝐿)
10		filunibas 23130	. . . . . . . 8 ⊢ (𝐿 ∈ (Fil‘𝑌) → ∪ 𝐿 = 𝑌)
11	10	ad2antlr 724	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ∪ 𝐿 = 𝑌)
12	9, 11	eqtrd 2776	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ∪ 𝑓 = 𝑌)
13	7, 12	oveq12d 7347	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → (∪ 𝑗 ↑m ∪ 𝑓) = (𝑋 ↑m 𝑌))
14	7	oveq1d 7344	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → (∪ 𝑗 FilMap 𝑔) = (𝑋 FilMap 𝑔))
15		simprr 770	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → 𝑓 = 𝐿)
16	14, 15	fveq12d 6826	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽 ∧ 𝑓 = 𝐿)) → ((∪ 𝑗 FilMap 𝑔)‘𝑓) = ((𝑋 FilMap 𝑔)‘𝐿))
17	3, 16	oveq12d 7347	. . . . 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 5180	. . . 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 22160	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
20	19	adantr 481	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐽 ∈ Top)
21		fvssunirn 6852	. . . . . 6 ⊢ (Fil‘𝑌) ⊆ ∪ ran Fil
22	21	sseli 3927	. . . . 5 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ ∪ ran Fil)
23	22	adantl 482	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐿 ∈ ∪ ran Fil)
24		ovex 7362	. . . . . 6 ⊢ (𝑋 ↑m 𝑌) ∈ V
25	24	mptex 7149	. . . . 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 7479	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fClusf 𝐿) = (𝑔 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿))))
28	27	3adant3 1131	. 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 485	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑔 = 𝐹) → 𝑔 = 𝐹)
30	29	oveq2d 7345	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑔 = 𝐹) → (𝑋 FilMap 𝑔) = (𝑋 FilMap 𝐹))
31	30	fveq1d 6821	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑔 = 𝐹) → ((𝑋 FilMap 𝑔)‘𝐿) = ((𝑋 FilMap 𝐹)‘𝐿))
32	31	oveq2d 7345	. 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 22173	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
34		filtop 23104	. . . 4 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝑌 ∈ 𝐿)
35		elmapg 8691	. . . 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 1469	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐹 ∈ (𝑋 ↑m 𝑌))
38		ovexd 7364	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)) ∈ V)
39	28, 32, 37, 38	fvmptd 6932	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 205   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  Vcvv 3441  ∪ cuni 4851   ↦ cmpt 5172  ran crn 5615  ⟶wf 6469  ‘cfv 6473  (class class class)co 7329   ∈ cmpo 7331   ↑_m cmap 8678  Topctop 22140  TopOnctopon 22157  Filcfil 23094   FilMap cfm 23182   fClus cfcls 23185   fClusf cfcf 23186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-ov 7332  df-oprab 7333  df-mpo 7334  df-map 8680  df-fbas 20692  df-top 22141  df-topon 22158  df-fil 23095  df-fcf 23191

This theorem is referenced by:  isfcf  23283  fcfelbas  23285  flfssfcf  23287  uffcfflf  23288  cnpfcfi  23289  cnpfcf  23290