Theorem acsfn1p 20734

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 5038	. . 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 3928	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → 𝑏 ∈ 𝑌)
4	3	biantrud 531	. . . . . . . 8 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → (𝑏 ∈ 𝑎 ↔ (𝑏 ∈ 𝑎 ∧ 𝑏 ∈ 𝑌)))
5		vex 3443	. . . . . . . . . 10 ⊢ 𝑏 ∈ V
6	5	snss 4740	. . . . . . . . 9 ⊢ (𝑏 ∈ 𝑎 ↔ {𝑏} ⊆ 𝑎)
7	6	bicomi 224	. . . . . . . 8 ⊢ ({𝑏} ⊆ 𝑎 ↔ 𝑏 ∈ 𝑎)
8		elin 3916	. . . . . . . 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 3079	. . . . 5 ⊢ (∀𝑏 ∈ (𝑋 ∩ 𝑌)({𝑏} ⊆ 𝑎 → 𝐸 ∈ 𝑎) ↔ ∀𝑏 ∈ (𝑋 ∩ 𝑌)(𝑏 ∈ (𝑎 ∩ 𝑌) → 𝐸 ∈ 𝑎))
12		elpwi 4560	. . . . . . . 8 ⊢ (𝑎 ∈ 𝒫 𝑋 → 𝑎 ⊆ 𝑋)
13	12	ssrind 4195	. . . . . . 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 4007	. . . . . 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 3404	. . 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 2782	. 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 17583	. . . 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 4001	. . . . . 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 767	. . . . . . . 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 4188	. . . . . . . . . . 11 ⊢ (𝑋 ∩ 𝑌) ⊆ 𝑋
27	26	sseli 3928	. . . . . . . . . 10 ⊢ (𝑏 ∈ (𝑋 ∩ 𝑌) → 𝑏 ∈ 𝑋)
28	27	ad2antlr 728	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → 𝑏 ∈ 𝑋)
29	28	snssd 4764	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → {𝑏} ⊆ 𝑋)
30		snfi 8982	. . . . . . . . 9 ⊢ {𝑏} ∈ Fin
31	30	a1i 11	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑏 ∈ (𝑋 ∩ 𝑌)) ∧ 𝐸 ∈ 𝑋) → {𝑏} ∈ Fin)
32		acsfn 17584	. . . . . . . 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 839	. . . . . . 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 3147	. . . . 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 17519	. . 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 585	. 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 2836	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 2114  ∀wral 3050  {crab 3398   ∩ cin 3899   ⊆ wss 3900  𝒫 cpw 4553  {csn 4579  ∩ ciin 4946  ‘cfv 6491  Fincfn 8885  Moorecmre 17503  ACScacs 17506

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-iin 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-om 7809  df-1o 8397  df-en 8886  df-fin 8889  df-mre 17507  df-mrc 17508  df-acs 17510

This theorem is referenced by: (None)