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Theorem fmid 22606
 Description: The filter map applied to the identity. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Mario Carneiro, 27-Aug-2015.)
Assertion
Ref Expression
fmid (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) = 𝐹)

Proof of Theorem fmid
Dummy variables 𝑡 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 filfbas 22494 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 f1oi 6636 . . . . 5 ( I ↾ 𝑋):𝑋1-1-onto𝑋
3 f1ofo 6606 . . . . 5 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋onto𝑋)
42, 3ax-mp 5 . . . 4 ( I ↾ 𝑋):𝑋onto𝑋
5 eqid 2798 . . . . 5 (𝑋filGen𝐹) = (𝑋filGen𝐹)
65elfm3 22596 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ ( I ↾ 𝑋):𝑋onto𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ ∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠)))
71, 4, 6sylancl 589 . . 3 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ ∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠)))
8 fgfil 22521 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
98rexeqdv 3366 . . 3 (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠)))
10 filelss 22498 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → 𝑠𝑋)
11 resiima 5915 . . . . . . . 8 (𝑠𝑋 → (( I ↾ 𝑋) “ 𝑠) = 𝑠)
1210, 11syl 17 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → (( I ↾ 𝑋) “ 𝑠) = 𝑠)
1312eqeq2d 2809 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡 = 𝑠))
14 equcom 2025 . . . . . 6 (𝑠 = 𝑡𝑡 = 𝑠)
1513, 14bitr4di 292 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑠 = 𝑡))
1615rexbidva 3256 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (∃𝑠𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠𝐹 𝑠 = 𝑡))
17 risset 3227 . . . 4 (𝑡𝐹 ↔ ∃𝑠𝐹 𝑠 = 𝑡)
1816, 17bitr4di 292 . . 3 (𝐹 ∈ (Fil‘𝑋) → (∃𝑠𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡𝐹))
197, 9, 183bitrd 308 . 2 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ 𝑡𝐹))
2019eqrdv 2796 1 (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107   ⊆ wss 3883   I cid 5428   ↾ cres 5525   “ cima 5526  –onto→wfo 6330  –1-1-onto→wf1o 6331  ‘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:  ufldom  22608
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