Theorem fmid 23019

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.

Step	Hyp	Ref	Expression
1		filfbas 22907	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2		f1oi 6737	. . . . 5 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋
3		f1ofo 6707	. . . . 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 2738	. . . . 5 ⊢ (𝑋filGen𝐹) = (𝑋filGen𝐹)
6	5	elfm3 23009	. . . 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 585	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ ∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠)))
8		fgfil 22934	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
9	8	rexeqdv 3340	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠 ∈ 𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠)))
10		filelss 22911	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ 𝑋)
11		resiima 5973	. . . . . . . 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 2749	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡 = 𝑠))
14		equcom 2022	. . . . . 6 ⊢ (𝑠 = 𝑡 ↔ 𝑡 = 𝑠)
15	13, 14	bitr4di 288	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑠 = 𝑡))
16	15	rexbidva 3224	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ 𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠 ∈ 𝐹 𝑠 = 𝑡))
17		risset 3193	. . . 4 ⊢ (𝑡 ∈ 𝐹 ↔ ∃𝑠 ∈ 𝐹 𝑠 = 𝑡)
18	16, 17	bitr4di 288	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ 𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡 ∈ 𝐹))
19	7, 9, 18	3bitrd 304	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ 𝑡 ∈ 𝐹))
20	19	eqrdv 2736	1 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) = 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∃wrex 3064   ⊆ wss 3883   I cid 5479   ↾ cres 5582   “ cima 5583  –onto→wfo 6416  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  fBascfbas 20498  filGencfg 20499  Filcfil 22904   FilMap cfm 22992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-fbas 20507  df-fg 20508  df-fil 22905  df-fm 22997

This theorem is referenced by:  ufldom  23021