Theorem fmid 23968

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 23856	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2		f1oi 6886	. . . . 5 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋
3		f1ofo 6855	. . . . 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 2737	. . . . 5 ⊢ (𝑋filGen𝐹) = (𝑋filGen𝐹)
6	5	elfm3 23958	. . . 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 586	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ ∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠)))
8		fgfil 23883	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
9	8	rexeqdv 3327	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ (𝑋filGen𝐹)𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠 ∈ 𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠)))
10		filelss 23860	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ 𝑋)
11		resiima 6094	. . . . . . . 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 2748	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡 = 𝑠))
14		equcom 2017	. . . . . 6 ⊢ (𝑠 = 𝑡 ↔ 𝑡 = 𝑠)
15	13, 14	bitr4di 289	. . . . 5 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑠 = 𝑡))
16	15	rexbidva 3177	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ 𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠 ∈ 𝐹 𝑠 = 𝑡))
17		risset 3233	. . . 4 ⊢ (𝑡 ∈ 𝐹 ↔ ∃𝑠 ∈ 𝐹 𝑠 = 𝑡)
18	16, 17	bitr4di 289	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ 𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡 ∈ 𝐹))
19	7, 9, 18	3bitrd 305	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) ↔ 𝑡 ∈ 𝐹))
20	19	eqrdv 2735	1 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) = 𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3070   ⊆ wss 3951   I cid 5577   ↾ cres 5687   “ cima 5688  –onto→wfo 6559  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431  fBascfbas 21352  filGencfg 21353  Filcfil 23853   FilMap cfm 23941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21361  df-fg 21362  df-fil 23854  df-fm 23946

This theorem is referenced by:  ufldom  23970