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Theorem fcfval 24057
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 23966 . . . . 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 771 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑗 = 𝐽)
43unieqd 4925 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑗 = 𝐽)
5 toponuni 22936 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
65ad2antrr 726 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑋 = 𝐽)
74, 6eqtr4d 2778 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑗 = 𝑋)
8 unieq 4923 . . . . . . . 8 (𝑓 = 𝐿 𝑓 = 𝐿)
98ad2antll 729 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑓 = 𝐿)
10 filunibas 23905 . . . . . . . 8 (𝐿 ∈ (Fil‘𝑌) → 𝐿 = 𝑌)
1110ad2antlr 727 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝐿 = 𝑌)
129, 11eqtrd 2775 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑓 = 𝑌)
137, 12oveq12d 7449 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → ( 𝑗m 𝑓) = (𝑋m 𝑌))
147oveq1d 7446 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → ( 𝑗 FilMap 𝑔) = (𝑋 FilMap 𝑔))
15 simprr 773 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑓 = 𝐿)
1614, 15fveq12d 6914 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → (( 𝑗 FilMap 𝑔)‘𝑓) = ((𝑋 FilMap 𝑔)‘𝐿))
173, 16oveq12d 7449 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → (𝑗 fClus (( 𝑗 FilMap 𝑔)‘𝑓)) = (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿)))
1813, 17mpteq12dv 5239 . . . 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 22935 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2019adantr 480 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐽 ∈ Top)
21 fvssunirn 6940 . . . . . 6 (Fil‘𝑌) ⊆ ran Fil
2221sseli 3991 . . . . 5 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ran Fil)
2322adantl 481 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐿 ran Fil)
24 ovex 7464 . . . . . 6 (𝑋m 𝑌) ∈ V
2524mptex 7243 . . . . 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, 26ovmpod 7585 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fClusf 𝐿) = (𝑔 ∈ (𝑋m 𝑌) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿))))
28273adant3 1131 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐽 fClusf 𝐿) = (𝑔 ∈ (𝑋m 𝑌) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿))))
29 simpr 484 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑔 = 𝐹) → 𝑔 = 𝐹)
3029oveq2d 7447 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑔 = 𝐹) → (𝑋 FilMap 𝑔) = (𝑋 FilMap 𝐹))
3130fveq1d 6909 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑔 = 𝐹) → ((𝑋 FilMap 𝑔)‘𝐿) = ((𝑋 FilMap 𝐹)‘𝐿))
3231oveq2d 7447 . 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 22948 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
34 filtop 23879 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝑌𝐿)
35 elmapg 8878 . . . 4 ((𝑋𝐽𝑌𝐿) → (𝐹 ∈ (𝑋m 𝑌) ↔ 𝐹:𝑌𝑋))
3633, 34, 35syl2an 596 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐹 ∈ (𝑋m 𝑌) ↔ 𝐹:𝑌𝑋))
3736biimp3ar 1469 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐹 ∈ (𝑋m 𝑌))
38 ovexd 7466 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)) ∈ V)
3928, 32, 37, 38fvmptd 7023 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478   cuni 4912  cmpt 5231  ran crn 5690  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  m cmap 8865  Topctop 22915  TopOnctopon 22932  Filcfil 23869   FilMap cfm 23957   fClus cfcls 23960   fClusf cfcf 23961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-fbas 21379  df-top 22916  df-topon 22933  df-fil 23870  df-fcf 23966
This theorem is referenced by:  isfcf  24058  fcfelbas  24060  flfssfcf  24062  uffcfflf  24063  cnpfcfi  24064  cnpfcf  24065
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