Theorem flfval 23980

Description: Given a function from a filtered set to a topological space, define the set of limit points of the function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Assertion

Ref	Expression
flfval	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))

Proof of Theorem flfval

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		toponmax 22916	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
2		filtop 23845	. . . . 5 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝑌 ∈ 𝐿)
3		elmapg 8783	. . . . 5 ⊢ ((𝑋 ∈ 𝐽 ∧ 𝑌 ∈ 𝐿) → (𝐹 ∈ (𝑋 ↑m 𝑌) ↔ 𝐹:𝑌⟶𝑋))
4	1, 2, 3	syl2an 602	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐹 ∈ (𝑋 ↑m 𝑌) ↔ 𝐹:𝑌⟶𝑋))
5	4	biimpar 478	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ 𝐹:𝑌⟶𝑋) → 𝐹 ∈ (𝑋 ↑m 𝑌))
6		flffval 23979	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))))
7	6	fveq1d 6836	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ((𝐽 fLimf 𝐿)‘𝐹) = ((𝑓 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))‘𝐹))
8		oveq2 7371	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑋 FilMap 𝑓) = (𝑋 FilMap 𝐹))
9	8	fveq1d 6836	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑋 FilMap 𝑓)‘𝐿) = ((𝑋 FilMap 𝐹)‘𝐿))
10	9	oveq2d 7379	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
11		eqid 2740	. . . . 5 ⊢ (𝑓 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))) = (𝑓 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))
12		ovex 7396	. . . . 5 ⊢ (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)) ∈ V
13	10, 11, 12	fvmpt 6942	. . . 4 ⊢ (𝐹 ∈ (𝑋 ↑m 𝑌) → ((𝑓 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
14	7, 13	sylan9eq 2795	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ 𝐹 ∈ (𝑋 ↑m 𝑌)) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
15	5, 14	syldan 597	. 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))
16	15	3impa 1115	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿)))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ↦ cmpt 5160  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363   ↑_m cmap 8770  TopOnctopon 22900  Filcfil 23835   FilMap cfm 23923   fLim cflim 23924   fLimf cflf 23925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-map 8772  df-fbas 21351  df-top 22884  df-topon 22901  df-fil 23836  df-flf 23930

This theorem is referenced by:  flfnei  23981  isflf  23983  hausflf  23987  flfcnp  23994  flfssfcf  24028  uffcfflf  24029  cnpfcf  24031  cnextcn  24057  tsmscls  24128  cnextucn  24292  cmetcaulem  25280  fmcncfil  34122