Theorem acsfn1p 19571

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 4969	. . 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 4156	. . . . . . . . . 10 ⊢ (𝑋 ∩ 𝑌) ⊆ 𝑌
3	2	sseli 3911	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → 𝑏 ∈ 𝑌)
4	3	biantrud 535	. . . . . . . 8 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → (𝑏 ∈ 𝑎 ↔ (𝑏 ∈ 𝑎 ∧ 𝑏 ∈ 𝑌)))
5		vex 3444	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
6	5	snss 4679	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝑎 ↔ {𝑏} ⊆ 𝑎)
7	6	bicomi 227	. . . . . . . 8 ⊢ ({𝑏} ⊆ 𝑎 ↔ 𝑏 ∈ 𝑎)
8		elin 3897	. . . . . . . 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 345	. . . . . 6 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → (({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎) ↔ (𝑏 ∈ (𝑎 ∩ 𝑌) → 𝐸 ∈ 𝑎)))
11	10	ralbiia 3132	. . . . 5 ⊢ (∀𝑏 ∈ (𝑋 ∩ 𝑌)({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎) ↔ ∀𝑏 ∈ (𝑋 ∩ 𝑌)(𝑏 ∈ (𝑎 ∩ 𝑌) → 𝐸 ∈ 𝑎))
12		elpwi 4506	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋)
13	12	ssrind 4162	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝑋 → (𝑎 ∩ 𝑌) ⊆ (𝑋 ∩ 𝑌))
14	13	adantl 485	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎 ∩ 𝑌) ⊆ (𝑋 ∩ 𝑌))
15		ralss 3985	. . . . . 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 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 3425	. . 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	syl5eq 2845	. 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 16921	. . . 4 ⊢ (𝑋 ∈ 𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))
21	20	adantr 484	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))
22		ssralv 3981	. . . . . 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 488	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → 𝐸 ∈ 𝑋)
26		inss1 4155	. . . . . . . . . . 11 ⊢ (𝑋 ∩ 𝑌) ⊆ 𝑋
27	26	sseli 3911	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → 𝑏 ∈ 𝑋)
28	27	ad2antlr 726	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → 𝑏 ∈ 𝑋)
29	28	snssd 4702	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → {𝑏} ⊆ 𝑋)
30		snfi 8577	. . . . . . . . 9 ⊢ {𝑏} ∈ Fin
31	30	a1i 11	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → {𝑏} ∈ Fin)
32		acsfn 16922	. . . . . . . 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 837	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋))
34	33	ex 416	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) → (𝐸 ∈ 𝑋 → {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋)))
35	34	ralimdva 3144	. . . . 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 410	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → ∀𝑏 ∈ (𝑋 ∩ 𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎)} ∈ (ACS‘𝑋))
38		mreriincl 16861	. . 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 587	. 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 2891	1 ⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑏 ∈ 𝑌 𝐸 ∈ 𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎 ∩ 𝑌)𝐸 ∈ 𝑎} ∈ (ACS‘𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2111  ∀wral 3106  {crab 3110   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497  {csn 4525  ∩ ciin 4882  ‘cfv 6324  Fincfn 8492  Moorecmre 16845  ACScacs 16848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-1o 8085  df-en 8493  df-fin 8496  df-mre 16849  df-mrc 16850  df-acs 16852

This theorem is referenced by: (None)