Theorem flffval 24107

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 23031	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		fvssunirn 6902	. . . 4 ⊢ (Fil‘𝑌) ⊆ ∪ ran Fil
3	2	sseli 3935	. . 3 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ ∪ ran Fil)
4		unieq 4879	. . . . . 6 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽)
5		unieq 4879	. . . . . 6 ⊢ (𝑦 = 𝐿 → ∪ 𝑦 = ∪ 𝐿)
6	4, 5	oveqan12d 7419	. . . . 5 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → (∪ 𝑥 ↑m ∪ 𝑦) = (∪ 𝐽 ↑m ∪ 𝐿))
7		simpl 487	. . . . . 6 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → 𝑥 = 𝐽)
8	4	adantr 485	. . . . . . . 8 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → ∪ 𝑥 = ∪ 𝐽)
9	8	oveq1d 7415	. . . . . . 7 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → (∪ 𝑥 FilMap 𝑓) = (∪ 𝐽 FilMap 𝑓))
10		simpr 489	. . . . . . 7 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → 𝑦 = 𝐿)
11	9, 10	fveq12d 6878	. . . . . 6 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → ((∪ 𝑥 FilMap 𝑓)‘𝑦) = ((∪ 𝐽 FilMap 𝑓)‘𝐿))
12	7, 11	oveq12d 7418	. . . . 5 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦)) = (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿)))
13	6, 12	mpteq12dv 5192	. . . 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 24058	. . . 4 ⊢ fLimf = (𝑥 ∈ Top, 𝑦 ∈ ∪ ran Fil ↦ (𝑓 ∈ (∪ 𝑥 ↑m ∪ 𝑦) ↦ (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦))))
15		ovex 7433	. . . . 5 ⊢ (∪ 𝐽 ↑m ∪ 𝐿) ∈ V
16	15	mptex 7211	. . . 4 ⊢ (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿) ↦ (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿))) ∈ V
17	13, 14, 16	ovmpoa 7555	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐿 ∈ ∪ ran Fil) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿) ↦ (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿))))
18	1, 3, 17	syl2an 607	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿) ↦ (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿))))
19		toponuni 23032	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
20	19	eqcomd 2771	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ 𝐽 = 𝑋)
21		filunibas 23999	. . . 4 ⊢ (𝐿 ∈ (Fil‘𝑌) → ∪ 𝐿 = 𝑌)
22	20, 21	oveqan12d 7419	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (∪ 𝐽 ↑m ∪ 𝐿) = (𝑋 ↑m 𝑌))
23	20	adantr 485	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ∪ 𝐽 = 𝑋)
24	23	oveq1d 7415	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (∪ 𝐽 FilMap 𝑓) = (𝑋 FilMap 𝑓))
25	24	fveq1d 6873	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ((∪ 𝐽 FilMap 𝑓)‘𝐿) = ((𝑋 FilMap 𝑓)‘𝐿))
26	25	oveq2d 7416	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿)) = (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))
27	22, 26	mpteq12dv 5192	. 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 2800	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 400   = wceq 1563   ∈ wcel 2145  ∪ cuni 4868   ↦ cmpt 5186  ran crn 5653  ‘cfv 6525  (class class class)co 7400   ↑_m cmap 8812  Topctop 23011  TopOnctopon 23028  Filcfil 23963   FilMap cfm 24051   fLim cflim 24052   fLimf cflf 24053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-fbas 21479  df-topon 23029  df-fil 23964  df-flf 24058

This theorem is referenced by:  flfval  24108