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Theorem fmid 21886
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 21774 . . . 4 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 f1oi 6287 . . . . 5 ( I ↾ 𝑋):𝑋1-1-onto𝑋
3 f1ofo 6257 . . . . 5 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋onto𝑋)
42, 3ax-mp 5 . . . 4 ( I ↾ 𝑋):𝑋onto𝑋
5 eqid 2724 . . . . 5 (𝑋filGen𝐹) = (𝑋filGen𝐹)
65elfm3 21876 . . . 4 ((𝐹 ∈ (fBas‘𝑋) ∧ ( I ↾ 𝑋):𝑋onto𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ ∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠)))
71, 4, 6sylancl 697 . . 3 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ ∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠)))
8 fgfil 21801 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
98rexeqdv 3248 . . 3 (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠)))
10 filelss 21778 . . . . . . . 8 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → 𝑠𝑋)
11 resiima 5590 . . . . . . . 8 (𝑠𝑋 → (( I ↾ 𝑋) “ 𝑠) = 𝑠)
1210, 11syl 17 . . . . . . 7 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → (( I ↾ 𝑋) “ 𝑠) = 𝑠)
1312eqeq2d 2734 . . . . . 6 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡 = 𝑠))
14 equcom 2064 . . . . . 6 (𝑠 = 𝑡𝑡 = 𝑠)
1513, 14syl6bbr 278 . . . . 5 ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑠 = 𝑡))
1615rexbidva 3151 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (∃𝑠𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠𝐹 𝑠 = 𝑡))
17 risset 3164 . . . 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 2722 1 (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  wrex 3015  wss 3680   I cid 5127  cres 5220  cima 5221  ontowfo 5999  1-1-ontowf1o 6000  cfv 6001  (class class class)co 6765  fBascfbas 19857  filGencfg 19858  Filcfil 21771   FilMap cfm 21859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-fbas 19866  df-fg 19867  df-fil 21772  df-fm 21864
This theorem is referenced by:  ufldom  21888
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