Theorem fmval 23859

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 23854	. . . . 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 5847	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4	3	fveq2d 6832	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (fBas‘dom 𝑓) = (fBas‘dom 𝐹))
5	4	adantl 481	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑓 = 𝐹) → (fBas‘dom 𝑓) = (fBas‘dom 𝐹))
6		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
7		imaeq1 6008	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓 “ 𝑦) = (𝐹 “ 𝑦))
8	7	mpteq2dv 5187	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)) = (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))
9	8	rneqd 5882	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)) = ran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))
10	6, 9	oveqan12d 7371	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑓 = 𝐹) → (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦))) = (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))
11	5, 10	mpteq12dv 5180	. . . . 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 6665	. . . . . . . 8 ⊢ (𝐹:𝑌⟶𝑋 → dom 𝐹 = 𝑌)
13	12	fveq2d 6832	. . . . . . 7 ⊢ (𝐹:𝑌⟶𝑋 → (fBas‘dom 𝐹) = (fBas‘𝑌))
14	13	mpteq1d 5183	. . . . . 6 ⊢ (𝐹:𝑌⟶𝑋 → (𝑏 ∈ (fBas‘dom 𝐹) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))))
15	14	3ad2ant3 1135	. . . . 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 2790	. . . 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 3458	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ V)
18	17	3ad2ant1 1133	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝑋 ∈ V)
19		simp3 1138	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐹:𝑌⟶𝑋)
20		elfvdm 6862	. . . . . 6 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas)
21	20	3ad2ant2 1134	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝑌 ∈ dom fBas)
22	19, 21	fexd 7167	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐹 ∈ V)
23		fvex 6841	. . . . . 6 ⊢ (fBas‘𝑌) ∈ V
24	23	mptex 7163	. . . . 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 7504	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))))
27	26	fveq1d 6830	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = ((𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))‘𝐵))
28		mpteq1 5182	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)) = (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)))
29	28	rneqd 5882	. . . . 5 ⊢ (𝑏 = 𝐵 → ran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)) = ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)))
30	29	oveq2d 7368	. . . 4 ⊢ (𝑏 = 𝐵 → (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))
31		eqid 2733	. . . 4 ⊢ (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))
32		ovex 7385	. . . 4 ⊢ (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))) ∈ V
33	30, 31, 32	fvmpt 6935	. . 3 ⊢ (𝐵 ∈ (fBas‘𝑌) → ((𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))
34	33	3ad2ant2 1134	. 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 2768	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ↦ cmpt 5174  dom cdm 5619  ran crn 5620   “ cima 5622  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352   ∈ cmpo 7354  fBascfbas 21281  filGencfg 21282   FilMap cfm 23849

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-pr 5372

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-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-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-fm 23854

This theorem is referenced by:  fmfil  23860  fmss  23862  elfm  23863  ucnextcn  24219  fmcfil  25200