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Theorem isacs1i 16365
Description: A closure system determined by a function is a closure system and algebraic. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
isacs1i ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (ACS‘𝑋))
Distinct variable groups:   𝐹,𝑠   𝑋,𝑠
Allowed substitution hint:   𝑉(𝑠)

Proof of Theorem isacs1i
Dummy variables 𝑎 𝑡 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3720 . . . 4 {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ⊆ 𝒫 𝑋
21a1i 11 . . 3 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ⊆ 𝒫 𝑋)
3 inss1 3866 . . . . . 6 (𝑋 𝑡) ⊆ 𝑋
4 elpw2g 4857 . . . . . 6 (𝑋𝑉 → ((𝑋 𝑡) ∈ 𝒫 𝑋 ↔ (𝑋 𝑡) ⊆ 𝑋))
53, 4mpbiri 248 . . . . 5 (𝑋𝑉 → (𝑋 𝑡) ∈ 𝒫 𝑋)
65ad2antrr 762 . . . 4 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝑋 𝑡) ∈ 𝒫 𝑋)
7 imassrn 5512 . . . . . . . . 9 (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ ran 𝐹
8 frn 6091 . . . . . . . . . 10 (𝐹:𝒫 𝑋⟶𝒫 𝑋 → ran 𝐹 ⊆ 𝒫 𝑋)
98adantl 481 . . . . . . . . 9 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → ran 𝐹 ⊆ 𝒫 𝑋)
107, 9syl5ss 3647 . . . . . . . 8 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝒫 𝑋)
1110unissd 4494 . . . . . . 7 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝒫 𝑋)
12 unipw 4948 . . . . . . 7 𝒫 𝑋 = 𝑋
1311, 12syl6sseq 3684 . . . . . 6 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑋)
1413adantr 480 . . . . 5 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑋)
15 inss2 3867 . . . . . . . . . . . . . 14 (𝑋 𝑡) ⊆ 𝑡
16 intss1 4524 . . . . . . . . . . . . . 14 (𝑎𝑡 𝑡𝑎)
1715, 16syl5ss 3647 . . . . . . . . . . . . 13 (𝑎𝑡 → (𝑋 𝑡) ⊆ 𝑎)
1817adantl 481 . . . . . . . . . . . 12 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → (𝑋 𝑡) ⊆ 𝑎)
19 sspwb 4947 . . . . . . . . . . . 12 ((𝑋 𝑡) ⊆ 𝑎 ↔ 𝒫 (𝑋 𝑡) ⊆ 𝒫 𝑎)
2018, 19sylib 208 . . . . . . . . . . 11 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → 𝒫 (𝑋 𝑡) ⊆ 𝒫 𝑎)
21 ssrin 3871 . . . . . . . . . . 11 (𝒫 (𝑋 𝑡) ⊆ 𝒫 𝑎 → (𝒫 (𝑋 𝑡) ∩ Fin) ⊆ (𝒫 𝑎 ∩ Fin))
2220, 21syl 17 . . . . . . . . . 10 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → (𝒫 (𝑋 𝑡) ∩ Fin) ⊆ (𝒫 𝑎 ∩ Fin))
23 imass2 5536 . . . . . . . . . 10 ((𝒫 (𝑋 𝑡) ∩ Fin) ⊆ (𝒫 𝑎 ∩ Fin) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ (𝐹 “ (𝒫 𝑎 ∩ Fin)))
2422, 23syl 17 . . . . . . . . 9 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ (𝐹 “ (𝒫 𝑎 ∩ Fin)))
2524unissd 4494 . . . . . . . 8 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ (𝐹 “ (𝒫 𝑎 ∩ Fin)))
26 ssel2 3631 . . . . . . . . . 10 ((𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∧ 𝑎𝑡) → 𝑎 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠})
27 pweq 4194 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑎 → 𝒫 𝑠 = 𝒫 𝑎)
2827ineq1d 3846 . . . . . . . . . . . . . . 15 (𝑠 = 𝑎 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑎 ∩ Fin))
2928imaeq2d 5501 . . . . . . . . . . . . . 14 (𝑠 = 𝑎 → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑎 ∩ Fin)))
3029unieqd 4478 . . . . . . . . . . . . 13 (𝑠 = 𝑎 (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑎 ∩ Fin)))
31 id 22 . . . . . . . . . . . . 13 (𝑠 = 𝑎𝑠 = 𝑎)
3230, 31sseq12d 3667 . . . . . . . . . . . 12 (𝑠 = 𝑎 → ( (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠 (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎))
3332elrab 3396 . . . . . . . . . . 11 (𝑎 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝑎 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎))
3433simprbi 479 . . . . . . . . . 10 (𝑎 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} → (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎)
3526, 34syl 17 . . . . . . . . 9 ((𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∧ 𝑎𝑡) → (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎)
3635adantll 750 . . . . . . . 8 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎)
3725, 36sstrd 3646 . . . . . . 7 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑎)
3837ralrimiva 2995 . . . . . 6 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → ∀𝑎𝑡 (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑎)
39 ssint 4525 . . . . . 6 ( (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑡 ↔ ∀𝑎𝑡 (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑎)
4038, 39sylibr 224 . . . . 5 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑡)
4114, 40ssind 3870 . . . 4 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ (𝑋 𝑡))
42 pweq 4194 . . . . . . . . 9 (𝑠 = (𝑋 𝑡) → 𝒫 𝑠 = 𝒫 (𝑋 𝑡))
4342ineq1d 3846 . . . . . . . 8 (𝑠 = (𝑋 𝑡) → (𝒫 𝑠 ∩ Fin) = (𝒫 (𝑋 𝑡) ∩ Fin))
4443imaeq2d 5501 . . . . . . 7 (𝑠 = (𝑋 𝑡) → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)))
4544unieqd 4478 . . . . . 6 (𝑠 = (𝑋 𝑡) → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)))
46 id 22 . . . . . 6 (𝑠 = (𝑋 𝑡) → 𝑠 = (𝑋 𝑡))
4745, 46sseq12d 3667 . . . . 5 (𝑠 = (𝑋 𝑡) → ( (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠 (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ (𝑋 𝑡)))
4847elrab 3396 . . . 4 ((𝑋 𝑡) ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ((𝑋 𝑡) ∈ 𝒫 𝑋 (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ (𝑋 𝑡)))
496, 41, 48sylanbrc 699 . . 3 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝑋 𝑡) ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠})
502, 49ismred2 16310 . 2 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (Moore‘𝑋))
51 fssxp 6098 . . . 4 (𝐹:𝒫 𝑋⟶𝒫 𝑋𝐹 ⊆ (𝒫 𝑋 × 𝒫 𝑋))
52 pwexg 4880 . . . . 5 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
53 xpexg 7002 . . . . 5 ((𝒫 𝑋 ∈ V ∧ 𝒫 𝑋 ∈ V) → (𝒫 𝑋 × 𝒫 𝑋) ∈ V)
5452, 52, 53syl2anc 694 . . . 4 (𝑋𝑉 → (𝒫 𝑋 × 𝒫 𝑋) ∈ V)
55 ssexg 4837 . . . 4 ((𝐹 ⊆ (𝒫 𝑋 × 𝒫 𝑋) ∧ (𝒫 𝑋 × 𝒫 𝑋) ∈ V) → 𝐹 ∈ V)
5651, 54, 55syl2anr 494 . . 3 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → 𝐹 ∈ V)
57 simpr 476 . . . 4 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → 𝐹:𝒫 𝑋⟶𝒫 𝑋)
58 pweq 4194 . . . . . . . . . 10 (𝑠 = 𝑡 → 𝒫 𝑠 = 𝒫 𝑡)
5958ineq1d 3846 . . . . . . . . 9 (𝑠 = 𝑡 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑡 ∩ Fin))
6059imaeq2d 5501 . . . . . . . 8 (𝑠 = 𝑡 → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑡 ∩ Fin)))
6160unieqd 4478 . . . . . . 7 (𝑠 = 𝑡 (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑡 ∩ Fin)))
62 id 22 . . . . . . 7 (𝑠 = 𝑡𝑠 = 𝑡)
6361, 62sseq12d 3667 . . . . . 6 (𝑠 = 𝑡 → ( (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠 (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
6463elrab3 3397 . . . . 5 (𝑡 ∈ 𝒫 𝑋 → (𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
6564rgen 2951 . . . 4 𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)
6657, 65jctir 560 . . 3 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → (𝐹:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)))
67 feq1 6064 . . . . 5 (𝑓 = 𝐹 → (𝑓:𝒫 𝑋⟶𝒫 𝑋𝐹:𝒫 𝑋⟶𝒫 𝑋))
68 imaeq1 5496 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 “ (𝒫 𝑡 ∩ Fin)) = (𝐹 “ (𝒫 𝑡 ∩ Fin)))
6968unieqd 4478 . . . . . . . 8 (𝑓 = 𝐹 (𝑓 “ (𝒫 𝑡 ∩ Fin)) = (𝐹 “ (𝒫 𝑡 ∩ Fin)))
7069sseq1d 3665 . . . . . . 7 (𝑓 = 𝐹 → ( (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡 (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
7170bibi2d 331 . . . . . 6 (𝑓 = 𝐹 → ((𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡) ↔ (𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)))
7271ralbidv 3015 . . . . 5 (𝑓 = 𝐹 → (∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡) ↔ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)))
7367, 72anbi12d 747 . . . 4 (𝑓 = 𝐹 → ((𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)) ↔ (𝐹:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))))
7473spcegv 3325 . . 3 (𝐹 ∈ V → ((𝐹:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)) → ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))))
7556, 66, 74sylc 65 . 2 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)))
76 isacs 16359 . 2 ({𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (ACS‘𝑋) ↔ ({𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))))
7750, 75, 76sylanbrc 699 1 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (ACS‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  cin 3606  wss 3607  𝒫 cpw 4191   cuni 4468   cint 4507   × cxp 5141  ran crn 5144  cima 5146  wf 5922  cfv 5926  Fincfn 7997  Moorecmre 16289  ACScacs 16292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-mre 16293  df-acs 16296
This theorem is referenced by:  acsfn  16367
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