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Mirrors > Home > MPE Home > Th. List > fmid | Structured version Visualization version GIF version |
Description: The filter map applied to the identity. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Mario Carneiro, 27-Aug-2015.) |
Ref | Expression |
---|---|
fmid | ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | filfbas 21873 | . . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | |
2 | f1oi 6316 | . . . . 5 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋 | |
3 | f1ofo 6286 | . . . . 5 ⊢ (( I ↾ 𝑋):𝑋–1-1-onto→𝑋 → ( I ↾ 𝑋):𝑋–onto→𝑋) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ ( I ↾ 𝑋):𝑋–onto→𝑋 |
5 | eqid 2771 | . . . . 5 ⊢ (𝑋filGen𝐹) = (𝑋filGen𝐹) | |
6 | 5 | elfm3 21975 | . . . 4 ⊢ ((𝐹 ∈ (fBas‘𝑋) ∧ ( I ↾ 𝑋):𝑋–onto→𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ ∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠))) |
7 | 1, 4, 6 | sylancl 568 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ ∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠))) |
8 | fgfil 21900 | . . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹) | |
9 | 8 | rexeqdv 3294 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠 ∈ 𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠))) |
10 | filelss 21877 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ 𝑋) | |
11 | resiima 5622 | . . . . . . . 8 ⊢ (𝑠 ⊆ 𝑋 → (( I ↾ 𝑋) “ 𝑠) = 𝑠) | |
12 | 10, 11 | syl 17 | . . . . . . 7 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → (( I ↾ 𝑋) “ 𝑠) = 𝑠) |
13 | 12 | eqeq2d 2781 | . . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡 = 𝑠)) |
14 | equcom 2103 | . . . . . 6 ⊢ (𝑠 = 𝑡 ↔ 𝑡 = 𝑠) | |
15 | 13, 14 | syl6bbr 278 | . . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑠 = 𝑡)) |
16 | 15 | rexbidva 3197 | . . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ 𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠 ∈ 𝐹 𝑠 = 𝑡)) |
17 | risset 3210 | . . . 4 ⊢ (𝑡 ∈ 𝐹 ↔ ∃𝑠 ∈ 𝐹 𝑠 = 𝑡) | |
18 | 16, 17 | syl6bbr 278 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ 𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡 ∈ 𝐹)) |
19 | 7, 9, 18 | 3bitrd 294 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ 𝑡 ∈ 𝐹)) |
20 | 19 | eqrdv 2769 | 1 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∃wrex 3062 ⊆ wss 3724 I cid 5157 ↾ cres 5252 “ cima 5253 –onto→wfo 6030 –1-1-onto→wf1o 6031 ‘cfv 6032 (class class class)co 6794 fBascfbas 19950 filGencfg 19951 Filcfil 21870 FilMap cfm 21958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7097 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 829 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3589 df-csb 3684 df-dif 3727 df-un 3729 df-in 3731 df-ss 3738 df-nul 4065 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5995 df-fun 6034 df-fn 6035 df-f 6036 df-f1 6037 df-fo 6038 df-f1o 6039 df-fv 6040 df-ov 6797 df-oprab 6798 df-mpt2 6799 df-fbas 19959 df-fg 19960 df-fil 21871 df-fm 21963 |
This theorem is referenced by: ufldom 21987 |
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