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Theorem fmid 21687
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 21575 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 f1oi 6136 . . . . 5 ( I ↾ 𝑋):𝑋1-1-onto𝑋
3 f1ofo 6106 . . . . 5 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋onto𝑋)
42, 3ax-mp 5 . . . 4 ( I ↾ 𝑋):𝑋onto𝑋
5 eqid 2621 . . . . 5 (𝑋filGen𝐹) = (𝑋filGen𝐹)
65elfm3 21677 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ ( I ↾ 𝑋):𝑋onto𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ ∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠)))
71, 4, 6sylancl 693 . . 3 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ ∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠)))
8 fgfil 21602 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
98rexeqdv 3137 . . 3 (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠)))
10 filelss 21579 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → 𝑠𝑋)
11 resiima 5444 . . . . . . . 8 (𝑠𝑋 → (( I ↾ 𝑋) “ 𝑠) = 𝑠)
1210, 11syl 17 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → (( I ↾ 𝑋) “ 𝑠) = 𝑠)
1312eqeq2d 2631 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡 = 𝑠))
14 equcom 1942 . . . . . 6 (𝑠 = 𝑡𝑡 = 𝑠)
1513, 14syl6bbr 278 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑠 = 𝑡))
1615rexbidva 3043 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (∃𝑠𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠𝐹 𝑠 = 𝑡))
17 risset 3056 . . . 4 (𝑡𝐹 ↔ ∃𝑠𝐹 𝑠 = 𝑡)
1816, 17syl6bbr 278 . . 3 (𝐹 ∈ (Fil‘𝑋) → (∃𝑠𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡𝐹))
197, 9, 183bitrd 294 . 2 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ 𝑡𝐹))
2019eqrdv 2619 1 (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2908  wss 3559   I cid 4989  cres 5081  cima 5082  ontowfo 5850  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  fBascfbas 19666  filGencfg 19667  Filcfil 21572   FilMap cfm 21660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-fbas 19675  df-fg 19676  df-fil 21573  df-fm 21665
This theorem is referenced by:  ufldom  21689
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