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Theorem fcfval 22641
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 22550 . . . . 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 770 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑗 = 𝐽)
43unieqd 4817 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑗 = 𝐽)
5 toponuni 21522 . . . . . . . 8 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
65ad2antrr 725 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑋 = 𝐽)
74, 6eqtr4d 2839 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑗 = 𝑋)
8 unieq 4814 . . . . . . . 8 (𝑓 = 𝐿 𝑓 = 𝐿)
98ad2antll 728 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑓 = 𝐿)
10 filunibas 22489 . . . . . . . 8 (𝐿 ∈ (Fil‘𝑌) → 𝐿 = 𝑌)
1110ad2antlr 726 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝐿 = 𝑌)
129, 11eqtrd 2836 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑓 = 𝑌)
137, 12oveq12d 7157 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → ( 𝑗m 𝑓) = (𝑋m 𝑌))
147oveq1d 7154 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → ( 𝑗 FilMap 𝑔) = (𝑋 FilMap 𝑔))
15 simprr 772 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → 𝑓 = 𝐿)
1614, 15fveq12d 6656 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → (( 𝑗 FilMap 𝑔)‘𝑓) = ((𝑋 FilMap 𝑔)‘𝐿))
173, 16oveq12d 7157 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ (𝑗 = 𝐽𝑓 = 𝐿)) → (𝑗 fClus (( 𝑗 FilMap 𝑔)‘𝑓)) = (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿)))
1813, 17mpteq12dv 5118 . . . 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 21521 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2019adantr 484 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐽 ∈ Top)
21 fvssunirn 6678 . . . . . 6 (Fil‘𝑌) ⊆ ran Fil
2221sseli 3914 . . . . 5 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ran Fil)
2322adantl 485 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → 𝐿 ran Fil)
24 ovex 7172 . . . . . 6 (𝑋m 𝑌) ∈ V
2524mptex 6967 . . . . 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 7285 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fClusf 𝐿) = (𝑔 ∈ (𝑋m 𝑌) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿))))
28273adant3 1129 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐽 fClusf 𝐿) = (𝑔 ∈ (𝑋m 𝑌) ↦ (𝐽 fClus ((𝑋 FilMap 𝑔)‘𝐿))))
29 simpr 488 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑔 = 𝐹) → 𝑔 = 𝐹)
3029oveq2d 7155 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑔 = 𝐹) → (𝑋 FilMap 𝑔) = (𝑋 FilMap 𝐹))
3130fveq1d 6651 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) ∧ 𝑔 = 𝐹) → ((𝑋 FilMap 𝑔)‘𝐿) = ((𝑋 FilMap 𝐹)‘𝐿))
3231oveq2d 7155 . 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 21534 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
34 filtop 22463 . . . 4 (𝐿 ∈ (Fil‘𝑌) → 𝑌𝐿)
35 elmapg 8406 . . . 4 ((𝑋𝐽𝑌𝐿) → (𝐹 ∈ (𝑋m 𝑌) ↔ 𝐹:𝑌𝑋))
3633, 34, 35syl2an 598 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐹 ∈ (𝑋m 𝑌) ↔ 𝐹:𝑌𝑋))
3736biimp3ar 1467 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → 𝐹 ∈ (𝑋m 𝑌))
38 ovexd 7174 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)) ∈ V)
3928, 32, 37, 38fvmptd 6756 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  Vcvv 3444   cuni 4803  cmpt 5113  ran crn 5524  wf 6324  cfv 6328  (class class class)co 7139  cmpo 7141  m cmap 8393  Topctop 21501  TopOnctopon 21518  Filcfil 22453   FilMap cfm 22541   fClus cfcls 22544   fClusf cfcf 22545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-fbas 20091  df-top 21502  df-topon 21519  df-fil 22454  df-fcf 22550
This theorem is referenced by:  isfcf  22642  fcfelbas  22644  flfssfcf  22646  uffcfflf  22647  cnpfcfi  22648  cnpfcf  22649
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