Theorem acsfn1p 20822

Description: Construction of a closure rule from a one-parameter partial operation. (Contributed by Stefan O'Rear, 12-Sep-2015.)

Assertion

Ref	Expression
acsfn1p	⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎 ∩ 𝑌)𝐸 ∈ 𝑎} ∈ (ACS‘𝑋))

Distinct variable groups:   𝑎,𝑏,𝑉   𝐸,𝑎   𝑋,𝑎,𝑏   𝑌,𝑎,𝑏

Allowed substitution hint:   𝐸(𝑏)

Proof of Theorem acsfn1p

Step	Hyp	Ref	Expression
1		riinrab 5107	. . 3 ⊢ (𝒫 𝑋 ∩ ∩ 𝑏 ∈ (𝑋 ∩ 𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)}) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑋 ∩ 𝑌)({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)}
2		inss2 4259	. . . . . . . . . 10 ⊢ (𝑋 ∩ 𝑌) ⊆ 𝑌
3	2	sseli 4004	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → 𝑏 ∈ 𝑌)
4	3	biantrud 531	. . . . . . . 8 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → (𝑏 ∈ 𝑎 ↔ (𝑏 ∈ 𝑎 ∧ 𝑏 ∈ 𝑌)))
5		vex 3492	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
6	5	snss 4810	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝑎 ↔ {𝑏} ⊆ 𝑎)
7	6	bicomi 224	. . . . . . . 8 ⊢ ({𝑏} ⊆ 𝑎 ↔ 𝑏 ∈ 𝑎)
8		elin 3992	. . . . . . . 8 ⊢ (𝑏 ∈ (𝑎 ∩ 𝑌) ↔ (𝑏 ∈ 𝑎 ∧ 𝑏 ∈ 𝑌))
9	4, 7, 8	3bitr4g 314	. . . . . . 7 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → ({𝑏} ⊆ 𝑎 ↔ 𝑏 ∈ (𝑎 ∩ 𝑌)))
10	9	imbi1d 341	. . . . . 6 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → (({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎) ↔ (𝑏 ∈ (𝑎 ∩ 𝑌) → 𝐸 ∈ 𝑎)))
11	10	ralbiia 3097	. . . . 5 ⊢ (∀𝑏 ∈ (𝑋 ∩ 𝑌)({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎) ↔ ∀𝑏 ∈ (𝑋 ∩ 𝑌)(𝑏 ∈ (𝑎 ∩ 𝑌) → 𝐸 ∈ 𝑎))
12		elpwi 4629	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋)
13	12	ssrind 4265	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∩ 𝑌) ⊆ (𝑋 ∩ 𝑌))
14	13	adantl 481	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∩ 𝑌) ⊆ (𝑋 ∩ 𝑌))
15		ralss 4083	. . . . . 6 ⊢ ((𝑎 ∩ 𝑌) ⊆ (𝑋 ∩ 𝑌) → (∀𝑏 ∈ (𝑎 ∩ 𝑌)𝐸 ∈ 𝑎 ↔ ∀𝑏 ∈ (𝑋 ∩ 𝑌)(𝑏 ∈ (𝑎 ∩ 𝑌) → 𝐸 ∈ 𝑎)))
16	14, 15	syl 17	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑏 ∈ (𝑎 ∩ 𝑌)𝐸 ∈ 𝑎 ↔ ∀𝑏 ∈ (𝑋 ∩ 𝑌)(𝑏 ∈ (𝑎 ∩ 𝑌) → 𝐸 ∈ 𝑎)))
17	11, 16	bitr4id 290	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑏 ∈ (𝑋 ∩ 𝑌)({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎) ↔ ∀𝑏 ∈ (𝑎 ∩ 𝑌)𝐸 ∈ 𝑎))
18	17	rabbidva 3450	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑋 ∩ 𝑌)({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎 ∩ 𝑌)𝐸 ∈ 𝑎})
19	1, 18	eqtrid 2792	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → (𝒫 𝑋 ∩ ∩ 𝑏 ∈ (𝑋 ∩ 𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)}) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎 ∩ 𝑌)𝐸 ∈ 𝑎})
20		mreacs 17716	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))
21	20	adantr 480	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))
22		ssralv 4077	. . . . . 6 ⊢ ((𝑋 ∩ 𝑌) ⊆ 𝑌 → (∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋 → ∀𝑏 ∈ (𝑋 ∩ 𝑌)𝐸 ∈ 𝑋))
23	2, 22	ax-mp 5	. . . . 5 ⊢ (∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋 → ∀𝑏 ∈ (𝑋 ∩ 𝑌)𝐸 ∈ 𝑋)
24		simpll 766	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → 𝑋 ∈ 𝑉)
25		simpr 484	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → 𝐸 ∈ 𝑋)
26		inss1 4258	. . . . . . . . . . 11 ⊢ (𝑋 ∩ 𝑌) ⊆ 𝑋
27	26	sseli 4004	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → 𝑏 ∈ 𝑋)
28	27	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → 𝑏 ∈ 𝑋)
29	28	snssd 4834	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → {𝑏} ⊆ 𝑋)
30		snfi 9109	. . . . . . . . 9 ⊢ {𝑏} ∈ Fin
31	30	a1i 11	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → {𝑏} ∈ Fin)
32		acsfn 17717	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐸 ∈ 𝑋) ∧ ({𝑏} ⊆ 𝑋 ∧ {𝑏} ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋))
33	24, 25, 29, 31, 32	syl22anc 838	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋))
34	33	ex 412	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) → (𝐸 ∈ 𝑋 → {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋)))
35	34	ralimdva 3173	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∀𝑏 ∈ (𝑋 ∩ 𝑌)𝐸 ∈ 𝑋 → ∀𝑏 ∈ (𝑋 ∩ 𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋)))
36	23, 35	syl5 34	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋 → ∀𝑏 ∈ (𝑋 ∩ 𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋)))
37	36	imp 406	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → ∀𝑏 ∈ (𝑋 ∩ 𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋))
38		mreriincl 17656	. . 3 ⊢ (((ACS‘𝑋) ∈ (Moore‘𝒫 𝑋) ∧ ∀𝑏 ∈ (𝑋 ∩ 𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋)) → (𝒫 𝑋 ∩ ∩ 𝑏 ∈ (𝑋 ∩ 𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)}) ∈ (ACS‘𝑋))
39	21, 37, 38	syl2anc 583	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → (𝒫 𝑋 ∩ ∩ 𝑏 ∈ (𝑋 ∩ 𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)}) ∈ (ACS‘𝑋))
40	19, 39	eqeltrrd 2845	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎 ∩ 𝑌)𝐸 ∈ 𝑎} ∈ (ACS‘𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  ∀wral 3067  {crab 3443   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  {csn 4648  ∩ ciin 5016  ‘cfv 6573  Fincfn 9003  Moorecmre 17640  ACScacs 17643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1o 8522  df-en 9004  df-fin 9007  df-mre 17644  df-mrc 17645  df-acs 17647

This theorem is referenced by: (None)