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Theorem fcfval 21747
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:𝑌𝑋) → ((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.
StepHypRef Expression
1 df-fcf 21656 . . . . 5 fClusf = (𝑗 ∈ Top, 𝑓 ran Fil ↦ (𝑔 ∈ ( 𝑗𝑚 𝑓) ↦ (𝑗 fClus (( 𝑗 FilMap 𝑔)‘𝑓))))
21a1i 11 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → fClusf = (𝑗 ∈ Top, 𝑓 ran Fil ↦ (𝑔 ∈ ( 𝑗𝑚 𝑓) ↦ (𝑗 fClus (( 𝑗 FilMap 𝑔)‘𝑓)))))
3 simprl 793 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑗 = 𝐽)
43unieqd 4412 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑗 = 𝐽)
5 toponuni 20642 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
65ad2antrr 761 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑋 = 𝐽)
74, 6eqtr4d 2658 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑗 = 𝑋)
8 unieq 4410 . . . . . . . 8 (𝑓 = 𝐿 𝑓 = 𝐿)
98ad2antll 764 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑓 = 𝐿)
10 filunibas 21595 . . . . . . . 8 (𝐿 ∈ (Fil‘𝑌) → 𝐿 = 𝑌)
1110ad2antlr 762 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝐿 = 𝑌)
129, 11eqtrd 2655 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑓 = 𝑌)
137, 12oveq12d 6622 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → ( 𝑗𝑚 𝑓) = (𝑋𝑚 𝑌))
147oveq1d 6619 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → ( 𝑗 FilMap 𝑔) = (𝑋 FilMap 𝑔))
15 simprr 795 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑓 = 𝐿)
1614, 15fveq12d 6154 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → (( 𝑗 FilMap 𝑔)‘𝑓) = ((𝑋 FilMap 𝑔)‘𝐿))
173, 16oveq12d 6622 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → (𝑗 fClus (( 𝑗 FilMap 𝑔)‘𝑓)) = (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿)))
1813, 17mpteq12dv 4693 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → (𝑔 ∈ ( 𝑗𝑚 𝑓) ↦ (𝑗 fClus (( 𝑗 FilMap 𝑔)‘𝑓))) = (𝑔 ∈ (𝑋𝑚 𝑌) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿))))
19 topontop 20641 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2019adantr 481 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐽 ∈ Top)
21 fvssunirn 6174 . . . . . 6 (Fil‘𝑌) ⊆ ran Fil
2221sseli 3579 . . . . 5 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ran Fil)
2322adantl 482 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐿 ran Fil)
24 ovex 6632 . . . . . 6 (𝑋𝑚 𝑌) ∈ V
2524mptex 6440 . . . . 5 (𝑔 ∈ (𝑋𝑚 𝑌) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿))) ∈ V
2625a1i 11 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝑔 ∈ (𝑋𝑚 𝑌) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿))) ∈ V)
272, 18, 20, 23, 26ovmpt2d 6741 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fClusf 𝐿) = (𝑔 ∈ (𝑋𝑚 𝑌) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿))))
28273adant3 1079 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐽 fClusf 𝐿) = (𝑔 ∈ (𝑋𝑚 𝑌) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿))))
29 simpr 477 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑔 = 𝐹) → 𝑔 = 𝐹)
3029oveq2d 6620 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑔 = 𝐹) → (𝑋 FilMap 𝑔) = (𝑋 FilMap 𝐹))
3130fveq1d 6150 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑔 = 𝐹) → ((𝑋 FilMap 𝑔)‘𝐿) = ((𝑋 FilMap 𝐹)‘𝐿))
3231oveq2d 6620 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑔 = 𝐹) → (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿)) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))
33 toponmax 20643 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
34 filtop 21569 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝑌𝐿)
35 elmapg 7815 . . . 4 ((𝑋𝐽𝑌𝐿) → (𝐹 ∈ (𝑋𝑚 𝑌) ↔ 𝐹:𝑌𝑋))
3633, 34, 35syl2an 494 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐹 ∈ (𝑋𝑚 𝑌) ↔ 𝐹:𝑌𝑋))
3736biimp3ar 1430 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐹 ∈ (𝑋𝑚 𝑌))
38 ovex 6632 . . 3 (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)) ∈ V
3938a1i 11 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)) ∈ V)
4028, 32, 37, 39fvmptd 6245 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3186   cuni 4402  cmpt 4673  ran crn 5075  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  𝑚 cmap 7802  Topctop 20617  TopOnctopon 20618  Filcfil 21559   FilMap cfm 21647   fClus cfcls 21650   fClusf cfcf 21651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-fbas 19662  df-top 20621  df-topon 20623  df-fil 21560  df-fcf 21656
This theorem is referenced by:  isfcf  21748  fcfelbas  21750  flfssfcf  21752  uffcfflf  21753  cnpfcfi  21754  cnpfcf  21755
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