Theorem acsfn1p 20881

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 5049	. . 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 4189	. . . . . . . . . 10 ⊢ (𝑋 ∩ 𝑌) ⊆ 𝑌
3	2	sseli 3932	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → 𝑏 ∈ 𝑌)
4	3	biantrud 540	. . . . . . . 8 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → (𝑏 ∈ 𝑎 ↔ (𝑏 ∈ 𝑎 ∧ 𝑏 ∈ 𝑌)))
5		vex 3457	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
6	5	snss 4749	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝑎 ↔ {𝑏} ⊆ 𝑎)
7	6	bicomi 227	. . . . . . . 8 ⊢ ({𝑏} ⊆ 𝑎 ↔ 𝑏 ∈ 𝑎)
8		elin 3920	. . . . . . . 8 ⊢ (𝑏 ∈ (𝑎 ∩ 𝑌) ↔ (𝑏 ∈ 𝑎 ∧ 𝑏 ∈ 𝑌))
9	4, 7, 8	3bitr4g 317	. . . . . . 7 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → ({𝑏} ⊆ 𝑎 ↔ 𝑏 ∈ (𝑎 ∩ 𝑌)))
10	9	imbi1d 344	. . . . . 6 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → (({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎) ↔ (𝑏 ∈ (𝑎 ∩ 𝑌) → 𝐸 ∈ 𝑎)))
11	10	ralbiia 3107	. . . . 5 ⊢ (∀𝑏 ∈ (𝑋 ∩ 𝑌)({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎) ↔ ∀𝑏 ∈ (𝑋 ∩ 𝑌)(𝑏 ∈ (𝑎 ∩ 𝑌) → 𝐸 ∈ 𝑎))
12		elpwi 4568	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋)
13	12	ssrind 4195	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∩ 𝑌) ⊆ (𝑋 ∩ 𝑌))
14	13	adantl 486	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∩ 𝑌) ⊆ (𝑋 ∩ 𝑌))
15		ralss 4009	. . . . . 6 ⊢ ((𝑎 ∩ 𝑌) ⊆ (𝑋 ∩ 𝑌) → (∀𝑏 ∈ (𝑎 ∩ 𝑌)𝐸 ∈ 𝑎 ↔ ∀𝑏 ∈ (𝑋 ∩ 𝑌)(𝑏 ∈ (𝑎 ∩ 𝑌) → 𝐸 ∈ 𝑎)))
16	14, 15	syl 18	. . . . 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 293	. . . 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 3420	. . 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 2808	. 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 17713	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))
21	20	adantr 485	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))
22		ssralv 4005	. . . . . 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 778	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → 𝑋 ∈ 𝑉)
25		simpr 489	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → 𝐸 ∈ 𝑋)
26		inss1 4188	. . . . . . . . . . 11 ⊢ (𝑋 ∩ 𝑌) ⊆ 𝑋
27	26	sseli 3932	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → 𝑏 ∈ 𝑋)
28	27	ad2antlr 739	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → 𝑏 ∈ 𝑋)
29	28	snssd 4751	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → {𝑏} ⊆ 𝑋)
30		snfi 9039	. . . . . . . . 9 ⊢ {𝑏} ∈ Fin
31	30	a1i 11	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → {𝑏} ∈ Fin)
32		acsfn 17714	. . . . . . . 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 851	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋))
34	33	ex 417	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) → (𝐸 ∈ 𝑋 → {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋)))
35	34	ralimdva 3175	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∀𝑏 ∈ (𝑋 ∩ 𝑌)𝐸 ∈ 𝑋 → ∀𝑏 ∈ (𝑋 ∩ 𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋)))
36	23, 35	syl5 35	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋 → ∀𝑏 ∈ (𝑋 ∩ 𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋)))
37	36	imp 411	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → ∀𝑏 ∈ (𝑋 ∩ 𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋))
38		mreriincl 17649	. . 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 595	. 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 2862	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎 ∩ 𝑌)𝐸 ∈ 𝑎} ∈ (ACS‘𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2141  ∀wral 3077  {crab 3414   ∩ cin 3903   ⊆ wss 3904  𝒫 cpw 4561  {csn 4588  ∩ ciin 4956  ‘cfv 6536  Fincfn 8942  Moorecmre 17633  ACScacs 17636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7862  df-1o 8452  df-en 8943  df-fin 8946  df-mre 17637  df-mrc 17638  df-acs 17640

This theorem is referenced by: (None)