Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fmfg Structured version   Visualization version   GIF version

Theorem fmfg 22595
 Description: The image filter of a filter base is the same as the image filter of its generated filter. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
elfm2.l 𝐿 = (𝑌filGen𝐵)
Assertion
Ref Expression
fmfg ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = ((𝑋 FilMap 𝐹)‘𝐿))

Proof of Theorem fmfg
Dummy variables 𝑠 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfm2.l . . . 4 𝐿 = (𝑌filGen𝐵)
21elfm2 22594 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
3 fgcl 22524 . . . . . 6 (𝐵 ∈ (fBas‘𝑌) → (𝑌filGen𝐵) ∈ (Fil‘𝑌))
41, 3eqeltrid 2894 . . . . 5 (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (Fil‘𝑌))
5 filfbas 22494 . . . . 5 (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
64, 5syl 17 . . . 4 (𝐵 ∈ (fBas‘𝑌) → 𝐿 ∈ (fBas‘𝑌))
7 elfm 22593 . . . 4 ((𝑋𝐶𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
86, 7syl3an2 1161 . . 3 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥𝑋 ∧ ∃𝑠𝐿 (𝐹𝑠) ⊆ 𝑥)))
92, 8bitr4d 285 . 2 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ 𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)))
109eqrdv 2796 1 ((𝑋𝐶𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = ((𝑋 FilMap 𝐹)‘𝐿))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3107   ⊆ wss 3883   “ cima 5526  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145  fBascfbas 20100  filGencfg 20101  Filcfil 22491   FilMap cfm 22579 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-fbas 20109  df-fg 20110  df-fil 22492  df-fm 22584 This theorem is referenced by:  fmfnfm  22604  cmetcaulem  23933
 Copyright terms: Public domain W3C validator