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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 21462 | . . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋)) | |
2 | f1oi 6086 | . . . . 5 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋 | |
3 | f1ofo 6057 | . . . . 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 2610 | . . . . 5 ⊢ (𝑋filGen𝐹) = (𝑋filGen𝐹) | |
6 | 5 | elfm3 21564 | . . . 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 693 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ ∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠))) |
8 | fgfil 21489 | . . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹) | |
9 | 8 | rexeqdv 3122 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠 ∈ 𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠))) |
10 | filelss 21466 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ 𝑋) | |
11 | resiima 5399 | . . . . . . . 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 2620 | . . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡 = 𝑠)) |
14 | equcom 1932 | . . . . . 6 ⊢ (𝑠 = 𝑡 ↔ 𝑡 = 𝑠) | |
15 | 13, 14 | syl6bbr 277 | . . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑠 = 𝑡)) |
16 | 15 | rexbidva 3031 | . . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ 𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠 ∈ 𝐹 𝑠 = 𝑡)) |
17 | risset 3044 | . . . 4 ⊢ (𝑡 ∈ 𝐹 ↔ ∃𝑠 ∈ 𝐹 𝑠 = 𝑡) | |
18 | 16, 17 | syl6bbr 277 | . . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ 𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡 ∈ 𝐹)) |
19 | 7, 9, 18 | 3bitrd 293 | . 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ 𝑡 ∈ 𝐹)) |
20 | 19 | eqrdv 2608 | 1 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ⊆ wss 3540 I cid 4948 ↾ cres 5040 “ cima 5041 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 fBascfbas 19555 filGencfg 19556 Filcfil 21459 FilMap cfm 21547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-fbas 19564 df-fg 19565 df-fil 21460 df-fm 21552 |
This theorem is referenced by: ufldom 21576 |
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