Theorem fmval 24317

Description: Introduce a function that takes a function from a filtered domain to a set and produces a filter which consists of supersets of images of filter elements. The functions which are dealt with by this function are similar to nets in topology. For example, suppose we have a sequence filtered by the filter generated by its tails under the usual positive integer ordering. Then the elements of this filter are precisely the supersets of tails of this sequence. Under this definition, it is not too difficult to see that the limit of a function in the filter sense captures the notion of convergence of a sequence. As a result, the notion of a filter generalizes many ideas associated with sequences, and this function is one way to make that relationship precise in Metamath. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Assertion

Ref	Expression
fmval	⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))

Distinct variable groups:   𝑦,𝐵   𝑦,𝐹   𝑦,𝑋   𝑦,𝑌   𝑦,𝐴

Proof of Theorem fmval

Dummy variables 𝑓 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fm 24312	. . . . 5 ⊢ FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)))))
2	1	a1i 11	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦))))))
3		dmeq 5897	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4	3	fveq2d 6920	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (fBas‘dom 𝑓) = (fBas‘dom 𝐹))
5	4	adantl 487	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑓 = 𝐹) → (fBas‘dom 𝑓) = (fBas‘dom 𝐹))
6		id 23	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
7		imaeq1 6065	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓 “ 𝑦) = (𝐹 “ 𝑦))
8	7	mpteq2dv 5205	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)) = (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))
9	8	rneqd 5932	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)) = ran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))
10	6, 9	oveqan12d 7470	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑓 = 𝐹) → (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦))) = (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))
11	5, 10	mpteq12dv 5198	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑓 = 𝐹) → (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)))) = (𝑏 ∈ (fBas‘dom 𝐹) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))))
12		fdm 6749	. . . . . . . 8 ⊢ (𝐹:𝑌⟶𝑋 → dom 𝐹 = 𝑌)
13	12	fveq2d 6920	. . . . . . 7 ⊢ (𝐹:𝑌⟶𝑋 → (fBas‘dom 𝐹) = (fBas‘𝑌))
14	13	mpteq1d 5201	. . . . . 6 ⊢ (𝐹:𝑌⟶𝑋 → (𝑏 ∈ (fBas‘dom 𝐹) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))))
15	14	3ad2ant3 1153	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑏 ∈ (fBas‘dom 𝐹) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))))
16	11, 15	sylan9eqr 2820	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝑥 = 𝑋 ∧ 𝑓 = 𝐹)) → (𝑏 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))))
17		elex 3475	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ V)
18	17	3ad2ant1 1151	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝑋 ∈ V)
19		simp3 1156	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐹:𝑌⟶𝑋)
20		elfvdm 6950	. . . . . 6 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas)
21	20	3ad2ant2 1152	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝑌 ∈ dom fBas)
22	19, 21	fexd 7264	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐹 ∈ V)
23		fvex 6929	. . . . . 6 ⊢ (fBas‘𝑌) ∈ V
24	23	mptex 7260	. . . . 5 ⊢ (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) ∈ V
25	24	a1i 11	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) ∈ V)
26	2, 16, 18, 22, 25	ovmpod 7604	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))))
27	26	fveq1d 6918	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = ((𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))‘𝐵))
28		mpteq1 5200	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)) = (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)))
29	28	rneqd 5932	. . . . 5 ⊢ (𝑏 = 𝐵 → ran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)) = ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)))
30	29	oveq2d 7467	. . . 4 ⊢ (𝑏 = 𝐵 → (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))
31		eqid 2763	. . . 4 ⊢ (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))
32		ovex 7484	. . . 4 ⊢ (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))) ∈ V
33	30, 31, 32	fvmpt 7024	. . 3 ⊢ (𝐵 ∈ (fBas‘𝑌) → ((𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))
34	33	3ad2ant2 1152	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))
35	27, 34	eqtrd 2798	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ↦ cmpt 5192  dom cdm 5658  ran crn 5659   “ cima 5661  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  fBascfbas 21717  filGencfg 21718   FilMap cfm 24307

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-fm 24312

This theorem is used by:  fmfil  24318  fmss  24320  elfm  24321  ucnextcn  24677  fmcfil  25648