Theorem flffval 24288

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 23211	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
2		fvssunirn 6908	. . . 4 ⊢ (Fil‘𝑌) ⊆ ∪ ran Fil
3	2	sseli 3927	. . 3 ⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ ∪ ran Fil)
4		unieq 4878	. . . . . 6 ⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪ 𝐽)
5		unieq 4878	. . . . . 6 ⊢ (𝑦 = 𝐿 → ∪ 𝑦 = ∪ 𝐿)
6	4, 5	oveqan12d 7431	. . . . 5 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → (∪ 𝑥 ↑m ∪ 𝑦) = (∪ 𝐽 ↑m ∪ 𝐿))
7		simpl 488	. . . . . 6 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → 𝑥 = 𝐽)
8	4	adantr 486	. . . . . . . 8 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → ∪ 𝑥 = ∪ 𝐽)
9	8	oveq1d 7427	. . . . . . 7 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → (∪ 𝑥 FilMap 𝑓) = (∪ 𝐽 FilMap 𝑓))
10		simpr 490	. . . . . . 7 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → 𝑦 = 𝐿)
11	9, 10	fveq12d 6884	. . . . . 6 ⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → ((∪ 𝑥 FilMap 𝑓)‘𝑦) = ((∪ 𝐽 FilMap 𝑓)‘𝐿))
12	7, 11	oveq12d 7430	. . . . 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 24239	. . . 4 ⊢ fLimf = (𝑥 ∈ Top, 𝑦 ∈ ∪ ran Fil ↦ (𝑓 ∈ (∪ 𝑥 ↑m ∪ 𝑦) ↦ (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦))))
15		ovex 7445	. . . . 5 ⊢ (∪ 𝐽 ↑m ∪ 𝐿) ∈ V
16	15	mptex 7221	. . . 4 ⊢ (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿) ↦ (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿))) ∈ V
17	13, 14, 16	ovmpoa 7567	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐿 ∈ ∪ ran Fil) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿) ↦ (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿))))
18	1, 3, 17	syl2an 608	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿) ↦ (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿))))
19		toponuni 23212	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
20	19	eqcomd 2767	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ 𝐽 = 𝑋)
21		filunibas 24180	. . . 4 ⊢ (𝐿 ∈ (Fil‘𝑌) → ∪ 𝐿 = 𝑌)
22	20, 21	oveqan12d 7431	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (∪ 𝐽 ↑m ∪ 𝐿) = (𝑋 ↑m 𝑌))
23	20	adantr 486	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ∪ 𝐽 = 𝑋)
24	23	oveq1d 7427	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (∪ 𝐽 FilMap 𝑓) = (𝑋 FilMap 𝑓))
25	24	fveq1d 6879	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ((∪ 𝐽 FilMap 𝑓)‘𝐿) = ((𝑋 FilMap 𝑓)‘𝐿))
26	25	oveq2d 7428	. . 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 2796	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∪ cuni 4867   ↦ cmpt 5186  ran crn 5652  ‘cfv 6531  (class class class)co 7412   ↑_m cmap 8831  Topctop 23191  TopOnctopon 23208  Filcfil 24144   FilMap cfm 24232   fLim cflim 24233   fLimf cflf 24234

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-fbas 21655  df-topon 23209  df-fil 24145  df-flf 24239

This theorem is used by:  flfval  24289