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Theorem acsfn1p 38543
Description: Construction of a closure rule from a one-parameter partial operation. (Contributed by Stefan O'Rear, 12-Sep-2015.)
Assertion
Ref Expression
acsfn1p ((𝑋𝑉 ∧ ∀𝑏𝑌 𝐸𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎𝑌)𝐸𝑎} ∈ (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
StepHypRef Expression
1 riinrab 4785 . . 3 (𝒫 𝑋 𝑏 ∈ (𝑋𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎𝐸𝑎)}) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑋𝑌)({𝑏} ⊆ 𝑎𝐸𝑎)}
2 elpwi 4358 . . . . . . . 8 (𝑎 ∈ 𝒫 𝑋𝑎𝑋)
32ssrind 4034 . . . . . . 7 (𝑎 ∈ 𝒫 𝑋 → (𝑎𝑌) ⊆ (𝑋𝑌))
43adantl 474 . . . . . 6 (((𝑋𝑉 ∧ ∀𝑏𝑌 𝐸𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑎𝑌) ⊆ (𝑋𝑌))
5 ralss 3863 . . . . . 6 ((𝑎𝑌) ⊆ (𝑋𝑌) → (∀𝑏 ∈ (𝑎𝑌)𝐸𝑎 ↔ ∀𝑏 ∈ (𝑋𝑌)(𝑏 ∈ (𝑎𝑌) → 𝐸𝑎)))
64, 5syl 17 . . . . 5 (((𝑋𝑉 ∧ ∀𝑏𝑌 𝐸𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑏 ∈ (𝑎𝑌)𝐸𝑎 ↔ ∀𝑏 ∈ (𝑋𝑌)(𝑏 ∈ (𝑎𝑌) → 𝐸𝑎)))
7 inss2 4028 . . . . . . . . . 10 (𝑋𝑌) ⊆ 𝑌
87sseli 3793 . . . . . . . . 9 (𝑏 ∈ (𝑋𝑌) → 𝑏𝑌)
98biantrud 528 . . . . . . . 8 (𝑏 ∈ (𝑋𝑌) → (𝑏𝑎 ↔ (𝑏𝑎𝑏𝑌)))
10 vex 3387 . . . . . . . . . 10 𝑏 ∈ V
1110snss 4503 . . . . . . . . 9 (𝑏𝑎 ↔ {𝑏} ⊆ 𝑎)
1211bicomi 216 . . . . . . . 8 ({𝑏} ⊆ 𝑎𝑏𝑎)
13 elin 3993 . . . . . . . 8 (𝑏 ∈ (𝑎𝑌) ↔ (𝑏𝑎𝑏𝑌))
149, 12, 133bitr4g 306 . . . . . . 7 (𝑏 ∈ (𝑋𝑌) → ({𝑏} ⊆ 𝑎𝑏 ∈ (𝑎𝑌)))
1514imbi1d 333 . . . . . 6 (𝑏 ∈ (𝑋𝑌) → (({𝑏} ⊆ 𝑎𝐸𝑎) ↔ (𝑏 ∈ (𝑎𝑌) → 𝐸𝑎)))
1615ralbiia 3159 . . . . 5 (∀𝑏 ∈ (𝑋𝑌)({𝑏} ⊆ 𝑎𝐸𝑎) ↔ ∀𝑏 ∈ (𝑋𝑌)(𝑏 ∈ (𝑎𝑌) → 𝐸𝑎))
176, 16syl6rbbr 282 . . . 4 (((𝑋𝑉 ∧ ∀𝑏𝑌 𝐸𝑋) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑏 ∈ (𝑋𝑌)({𝑏} ⊆ 𝑎𝐸𝑎) ↔ ∀𝑏 ∈ (𝑎𝑌)𝐸𝑎))
1817rabbidva 3371 . . 3 ((𝑋𝑉 ∧ ∀𝑏𝑌 𝐸𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑋𝑌)({𝑏} ⊆ 𝑎𝐸𝑎)} = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎𝑌)𝐸𝑎})
191, 18syl5eq 2844 . 2 ((𝑋𝑉 ∧ ∀𝑏𝑌 𝐸𝑋) → (𝒫 𝑋 𝑏 ∈ (𝑋𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎𝐸𝑎)}) = {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎𝑌)𝐸𝑎})
20 mreacs 16630 . . . 4 (𝑋𝑉 → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))
2120adantr 473 . . 3 ((𝑋𝑉 ∧ ∀𝑏𝑌 𝐸𝑋) → (ACS‘𝑋) ∈ (Moore‘𝒫 𝑋))
22 ssralv 3861 . . . . . 6 ((𝑋𝑌) ⊆ 𝑌 → (∀𝑏𝑌 𝐸𝑋 → ∀𝑏 ∈ (𝑋𝑌)𝐸𝑋))
237, 22ax-mp 5 . . . . 5 (∀𝑏𝑌 𝐸𝑋 → ∀𝑏 ∈ (𝑋𝑌)𝐸𝑋)
24 simpll 784 . . . . . . . 8 (((𝑋𝑉𝑏 ∈ (𝑋𝑌)) ∧ 𝐸𝑋) → 𝑋𝑉)
25 simpr 478 . . . . . . . 8 (((𝑋𝑉𝑏 ∈ (𝑋𝑌)) ∧ 𝐸𝑋) → 𝐸𝑋)
26 inss1 4027 . . . . . . . . . . 11 (𝑋𝑌) ⊆ 𝑋
2726sseli 3793 . . . . . . . . . 10 (𝑏 ∈ (𝑋𝑌) → 𝑏𝑋)
2827ad2antlr 719 . . . . . . . . 9 (((𝑋𝑉𝑏 ∈ (𝑋𝑌)) ∧ 𝐸𝑋) → 𝑏𝑋)
2928snssd 4527 . . . . . . . 8 (((𝑋𝑉𝑏 ∈ (𝑋𝑌)) ∧ 𝐸𝑋) → {𝑏} ⊆ 𝑋)
30 snfi 8279 . . . . . . . . 9 {𝑏} ∈ Fin
3130a1i 11 . . . . . . . 8 (((𝑋𝑉𝑏 ∈ (𝑋𝑌)) ∧ 𝐸𝑋) → {𝑏} ∈ Fin)
32 acsfn 16631 . . . . . . . 8 (((𝑋𝑉𝐸𝑋) ∧ ({𝑏} ⊆ 𝑋 ∧ {𝑏} ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎𝐸𝑎)} ∈ (ACS‘𝑋))
3324, 25, 29, 31, 32syl22anc 868 . . . . . . 7 (((𝑋𝑉𝑏 ∈ (𝑋𝑌)) ∧ 𝐸𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎𝐸𝑎)} ∈ (ACS‘𝑋))
3433ex 402 . . . . . 6 ((𝑋𝑉𝑏 ∈ (𝑋𝑌)) → (𝐸𝑋 → {𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎𝐸𝑎)} ∈ (ACS‘𝑋)))
3534ralimdva 3142 . . . . 5 (𝑋𝑉 → (∀𝑏 ∈ (𝑋𝑌)𝐸𝑋 → ∀𝑏 ∈ (𝑋𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎𝐸𝑎)} ∈ (ACS‘𝑋)))
3623, 35syl5 34 . . . 4 (𝑋𝑉 → (∀𝑏𝑌 𝐸𝑋 → ∀𝑏 ∈ (𝑋𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎𝐸𝑎)} ∈ (ACS‘𝑋)))
3736imp 396 . . 3 ((𝑋𝑉 ∧ ∀𝑏𝑌 𝐸𝑋) → ∀𝑏 ∈ (𝑋𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎𝐸𝑎)} ∈ (ACS‘𝑋))
38 mreriincl 16570 . . 3 (((ACS‘𝑋) ∈ (Moore‘𝒫 𝑋) ∧ ∀𝑏 ∈ (𝑋𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎𝐸𝑎)} ∈ (ACS‘𝑋)) → (𝒫 𝑋 𝑏 ∈ (𝑋𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎𝐸𝑎)}) ∈ (ACS‘𝑋))
3921, 37, 38syl2anc 580 . 2 ((𝑋𝑉 ∧ ∀𝑏𝑌 𝐸𝑋) → (𝒫 𝑋 𝑏 ∈ (𝑋𝑌){𝑎 ∈ 𝒫 𝑋 ∣ ({𝑏} ⊆ 𝑎𝐸𝑎)}) ∈ (ACS‘𝑋))
4019, 39eqeltrrd 2878 1 ((𝑋𝑉 ∧ ∀𝑏𝑌 𝐸𝑋) → {𝑎 ∈ 𝒫 𝑋 ∣ ∀𝑏 ∈ (𝑎𝑌)𝐸𝑎} ∈ (ACS‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wcel 2157  wral 3088  {crab 3092  cin 3767  wss 3768  𝒫 cpw 4348  {csn 4367   ciin 4710  cfv 6100  Fincfn 8194  Moorecmre 16554  ACScacs 16557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pow 5034  ax-pr 5096  ax-un 7182
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3386  df-sbc 3633  df-csb 3728  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-pss 3784  df-nul 4115  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4628  df-int 4667  df-iun 4711  df-iin 4712  df-br 4843  df-opab 4905  df-mpt 4922  df-tr 4945  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-ord 5943  df-on 5944  df-lim 5945  df-suc 5946  df-iota 6063  df-fun 6102  df-fn 6103  df-f 6104  df-f1 6105  df-fo 6106  df-f1o 6107  df-fv 6108  df-om 7299  df-1o 7798  df-en 8195  df-fin 8198  df-mre 16558  df-mrc 16559  df-acs 16561
This theorem is referenced by: (None)
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