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Theorem fmid 23984
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 23872 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 f1oi 6887 . . . . 5 ( I ↾ 𝑋):𝑋1-1-onto𝑋
3 f1ofo 6856 . . . . 5 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋onto𝑋)
42, 3ax-mp 5 . . . 4 ( I ↾ 𝑋):𝑋onto𝑋
5 eqid 2735 . . . . 5 (𝑋filGen𝐹) = (𝑋filGen𝐹)
65elfm3 23974 . . . 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 23899 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
98rexeqdv 3325 . . 3 (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠)))
10 filelss 23876 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → 𝑠𝑋)
11 resiima 6096 . . . . . . . 8 (𝑠𝑋 → (( I ↾ 𝑋) “ 𝑠) = 𝑠)
1210, 11syl 17 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → (( I ↾ 𝑋) “ 𝑠) = 𝑠)
1312eqeq2d 2746 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡 = 𝑠))
14 equcom 2015 . . . . . 6 (𝑠 = 𝑡𝑡 = 𝑠)
1513, 14bitr4di 289 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑠 = 𝑡))
1615rexbidva 3175 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (∃𝑠𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠𝐹 𝑠 = 𝑡))
17 risset 3231 . . . 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 2733 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 1537  wcel 2106  wrex 3068  wss 3963   I cid 5582  cres 5691  cima 5692  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  fBascfbas 21370  filGencfg 21371  Filcfil 23869   FilMap cfm 23957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21379  df-fg 21380  df-fil 23870  df-fm 23962
This theorem is referenced by:  ufldom  23986
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