Theorem fmval 23092

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 23087	. . . . 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 5811	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4	3	fveq2d 6775	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (fBas‘dom 𝑓) = (fBas‘dom 𝐹))
5	4	adantl 482	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑓 = 𝐹) → (fBas‘dom 𝑓) = (fBas‘dom 𝐹))
6		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
7		imaeq1 5963	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓 “ 𝑦) = (𝐹 “ 𝑦))
8	7	mpteq2dv 5181	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)) = (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))
9	8	rneqd 5846	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦)) = ran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))
10	6, 9	oveqan12d 7290	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑓 = 𝐹) → (𝑥filGenran (𝑦 ∈ 𝑏 ↦ (𝑓 “ 𝑦))) = (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))
11	5, 10	mpteq12dv 5170	. . . . 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 6607	. . . . . . . 8 ⊢ (𝐹:𝑌⟶𝑋 → dom 𝐹 = 𝑌)
13	12	fveq2d 6775	. . . . . . 7 ⊢ (𝐹:𝑌⟶𝑋 → (fBas‘dom 𝐹) = (fBas‘𝑌))
14	13	mpteq1d 5174	. . . . . 6 ⊢ (𝐹:𝑌⟶𝑋 → (𝑏 ∈ (fBas‘dom 𝐹) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))))
15	14	3ad2ant3 1134	. . . . 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 2802	. . . 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 3449	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → 𝑋 ∈ V)
18	17	3ad2ant1 1132	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝑋 ∈ V)
19		simp3 1137	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐹:𝑌⟶𝑋)
20		elfvdm 6803	. . . . . 6 ⊢ (𝐵 ∈ (fBas‘𝑌) → 𝑌 ∈ dom fBas)
21	20	3ad2ant2 1133	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝑌 ∈ dom fBas)
22	19, 21	fexd 7100	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐹 ∈ V)
23		fvex 6784	. . . . . 6 ⊢ (fBas‘𝑌) ∈ V
24	23	mptex 7096	. . . . 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 7419	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))))
27	26	fveq1d 6773	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = ((𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))‘𝐵))
28		mpteq1 5172	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)) = (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)))
29	28	rneqd 5846	. . . . 5 ⊢ (𝑏 = 𝐵 → ran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)) = ran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)))
30	29	oveq2d 7287	. . . 4 ⊢ (𝑏 = 𝐵 → (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))
31		eqid 2740	. . . 4 ⊢ (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦)))) = (𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))
32		ovex 7304	. . . 4 ⊢ (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))) ∈ V
33	30, 31, 32	fvmpt 6872	. . 3 ⊢ (𝐵 ∈ (fBas‘𝑌) → ((𝑏 ∈ (fBas‘𝑌) ↦ (𝑋filGenran (𝑦 ∈ 𝑏 ↦ (𝐹 “ 𝑦))))‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))
34	33	3ad2ant2 1133	. 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 2780	1 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦))))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1542   ∈ wcel 2110  Vcvv 3431   ↦ cmpt 5162  dom cdm 5590  ran crn 5591   “ cima 5593  ⟶wf 6428  ‘cfv 6432  (class class class)co 7271   ∈ cmpo 7273  fBascfbas 20583  filGencfg 20584   FilMap cfm 23082

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-fm 23087

This theorem is referenced by:  fmfil  23093  fmss  23095  elfm  23096  ucnextcn  23454  fmcfil  24434