Theorem fmid 23876

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 23764	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2		f1oi 6806	. . . . 5 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋
3		f1ofo 6775	. . . . 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 2733	. . . . 5 ⊢ (𝑋filGen𝐹) = (𝑋filGen𝐹)
6	5	elfm3 23866	. . . 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 23791	. . . 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 23768	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ 𝑋)
11		resiima 6029	. . . . . . . 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 2744	. . . . . 6 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → (𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ 𝑡 = 𝑠))
14		equcom 2019	. . . . . 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 3155	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑠 ∈ 𝐹 𝑡 = (( I ↾ 𝑋) “ 𝑠) ↔ ∃𝑠 ∈ 𝐹 𝑠 = 𝑡))
17		risset 3208	. . . 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 2731	1 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) = 𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3057   ⊆ wss 3898   I cid 5513   ↾ cres 5621   “ cima 5622  –onto→wfo 6484  –1-1-onto→wf1o 6485  ‘cfv 6486  (class class class)co 7352  fBascfbas 21281  filGencfg 21282  Filcfil 23761   FilMap cfm 23849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-fbas 21290  df-fg 21291  df-fil 23762  df-fm 23854

This theorem is referenced by:  ufldom  23878