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

Theorem fmid 23968
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 23856 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 f1oi 6886 . . . . 5 ( I ↾ 𝑋):𝑋1-1-onto𝑋
3 f1ofo 6855 . . . . 5 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋onto𝑋)
42, 3ax-mp 5 . . . 4 ( I ↾ 𝑋):𝑋onto𝑋
5 eqid 2737 . . . . 5 (𝑋filGen𝐹) = (𝑋filGen𝐹)
65elfm3 23958 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ ( I ↾ 𝑋):𝑋onto𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ ∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠)))
71, 4, 6sylancl 586 . . 3 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ ∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠)))
8 fgfil 23883 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
98rexeqdv 3327 . . 3 (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠)))
10 filelss 23860 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → 𝑠𝑋)
11 resiima 6094 . . . . . . . 8 (𝑠𝑋 → (( I ↾ 𝑋) “ 𝑠) = 𝑠)
1210, 11syl 17 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → (( I ↾ 𝑋) “ 𝑠) = 𝑠)
1312eqeq2d 2748 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡 = 𝑠))
14 equcom 2017 . . . . . 6 (𝑠 = 𝑡𝑡 = 𝑠)
1513, 14bitr4di 289 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑠 = 𝑡))
1615rexbidva 3177 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (∃𝑠𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠𝐹 𝑠 = 𝑡))
17 risset 3233 . . . 4 (𝑡𝐹 ↔ ∃𝑠𝐹 𝑠 = 𝑡)
1816, 17bitr4di 289 . . 3 (𝐹 ∈ (Fil‘𝑋) → (∃𝑠𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡𝐹))
197, 9, 183bitrd 305 . 2 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ 𝑡𝐹))
2019eqrdv 2735 1 (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  wss 3951   I cid 5577  cres 5687  cima 5688  ontowfo 6559  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  fBascfbas 21352  filGencfg 21353  Filcfil 23853   FilMap cfm 23941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21361  df-fg 21362  df-fil 23854  df-fm 23946
This theorem is referenced by:  ufldom  23970
  Copyright terms: Public domain W3C validator