Theorem flffval 23905

Description: Given a topology and a filtered set, return the convergence function on the functions from the filtered set to the base set of the topological space. (Contributed by Jeff Hankins, 14-Oct-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Assertion

Ref	Expression
flffval	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))))

Distinct variable groups:   𝑓,𝐽   𝑓,𝑋   𝑓,𝑌   𝑓,𝐿

Proof of Theorem flffval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		topontop 22829	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		fvssunirn 6859	. . . 4 ⊢ (Fil‘𝑌) ⊆ ∪ ran Fil
3	2	sseli 3926	. . 3 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ ∪ ran Fil)
4		unieq 4869	. . . . . 6 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽)
5		unieq 4869	. . . . . 6 ⊢ (𝑦 = 𝐿 → ∪ 𝑦 = ∪ 𝐿)
6	4, 5	oveqan12d 7371	. . . . 5 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → (∪ 𝑥 ↑m ∪ 𝑦) = (∪ 𝐽 ↑m ∪ 𝐿))
7		simpl 482	. . . . . 6 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → 𝑥 = 𝐽)
8	4	adantr 480	. . . . . . . 8 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → ∪ 𝑥 = ∪ 𝐽)
9	8	oveq1d 7367	. . . . . . 7 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → (∪ 𝑥 FilMap 𝑓) = (∪ 𝐽 FilMap 𝑓))
10		simpr 484	. . . . . . 7 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → 𝑦 = 𝐿)
11	9, 10	fveq12d 6835	. . . . . 6 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → ((∪ 𝑥 FilMap 𝑓)‘𝑦) = ((∪ 𝐽 FilMap 𝑓)‘𝐿))
12	7, 11	oveq12d 7370	. . . . 5 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦)) = (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿)))
13	6, 12	mpteq12dv 5180	. . . 4 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → (𝑓 ∈ (∪ 𝑥 ↑m ∪ 𝑦) ↦ (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦))) = (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿) ↦ (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿))))
14		df-flf 23856	. . . 4 ⊢ fLimf = (𝑥 ∈ Top, 𝑦 ∈ ∪ ran Fil ↦ (𝑓 ∈ (∪ 𝑥 ↑m ∪ 𝑦) ↦ (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦))))
15		ovex 7385	. . . . 5 ⊢ (∪ 𝐽 ↑m ∪ 𝐿) ∈ V
16	15	mptex 7163	. . . 4 ⊢ (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿) ↦ (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿))) ∈ V
17	13, 14, 16	ovmpoa 7507	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐿 ∈ ∪ ran Fil) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿) ↦ (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿))))
18	1, 3, 17	syl2an 596	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿) ↦ (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿))))
19		toponuni 22830	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
20	19	eqcomd 2739	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ 𝐽 = 𝑋)
21		filunibas 23797	. . . 4 ⊢ (𝐿 ∈ (Fil‘𝑌) → ∪ 𝐿 = 𝑌)
22	20, 21	oveqan12d 7371	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (∪ 𝐽 ↑m ∪ 𝐿) = (𝑋 ↑m 𝑌))
23	20	adantr 480	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ∪ 𝐽 = 𝑋)
24	23	oveq1d 7367	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (∪ 𝐽 FilMap 𝑓) = (𝑋 FilMap 𝑓))
25	24	fveq1d 6830	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ((∪ 𝐽 FilMap 𝑓)‘𝐿) = ((𝑋 FilMap 𝑓)‘𝐿))
26	25	oveq2d 7368	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿)) = (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))
27	22, 26	mpteq12dv 5180	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿) ↦ (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿))) = (𝑓 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))))
28	18, 27	eqtrd 2768	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∪ cuni 4858   ↦ cmpt 5174  ran crn 5620  ‘cfv 6486  (class class class)co 7352   ↑_m cmap 8756  Topctop 22809  TopOnctopon 22826  Filcfil 23761   FilMap cfm 23849   fLim cflim 23850   fLimf cflf 23851

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-fbas 21290  df-topon 22827  df-fil 23762  df-flf 23856

This theorem is referenced by:  flfval  23906