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Theorem isacs1i 16677
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 3914 . . . 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 4059 . . . . . 6 (𝑋 𝑡) ⊆ 𝑋
4 elpw2g 5051 . . . . . 6 (𝑋𝑉 → ((𝑋 𝑡) ∈ 𝒫 𝑋 ↔ (𝑋 𝑡) ⊆ 𝑋))
53, 4mpbiri 250 . . . . 5 (𝑋𝑉 → (𝑋 𝑡) ∈ 𝒫 𝑋)
65ad2antrr 717 . . . 4 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝑋 𝑡) ∈ 𝒫 𝑋)
7 imassrn 5722 . . . . . . . . 9 (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ ran 𝐹
8 frn 6288 . . . . . . . . . 10 (𝐹:𝒫 𝑋⟶𝒫 𝑋 → ran 𝐹 ⊆ 𝒫 𝑋)
98adantl 475 . . . . . . . . 9 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → ran 𝐹 ⊆ 𝒫 𝑋)
107, 9syl5ss 3838 . . . . . . . 8 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝒫 𝑋)
1110unissd 4686 . . . . . . 7 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝒫 𝑋)
12 unipw 5141 . . . . . . 7 𝒫 𝑋 = 𝑋
1311, 12syl6sseq 3876 . . . . . 6 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑋)
1413adantr 474 . . . . 5 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑋)
15 inss2 4060 . . . . . . . . . . . . . 14 (𝑋 𝑡) ⊆ 𝑡
16 intss1 4714 . . . . . . . . . . . . . 14 (𝑎𝑡 𝑡𝑎)
1715, 16syl5ss 3838 . . . . . . . . . . . . 13 (𝑎𝑡 → (𝑋 𝑡) ⊆ 𝑎)
1817adantl 475 . . . . . . . . . . . 12 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → (𝑋 𝑡) ⊆ 𝑎)
19 sspwb 5140 . . . . . . . . . . . 12 ((𝑋 𝑡) ⊆ 𝑎 ↔ 𝒫 (𝑋 𝑡) ⊆ 𝒫 𝑎)
2018, 19sylib 210 . . . . . . . . . . 11 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → 𝒫 (𝑋 𝑡) ⊆ 𝒫 𝑎)
2120ssrind 4066 . . . . . . . . . 10 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → (𝒫 (𝑋 𝑡) ∩ Fin) ⊆ (𝒫 𝑎 ∩ Fin))
22 imass2 5746 . . . . . . . . . 10 ((𝒫 (𝑋 𝑡) ∩ Fin) ⊆ (𝒫 𝑎 ∩ Fin) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ (𝐹 “ (𝒫 𝑎 ∩ Fin)))
2321, 22syl 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)))
2423unissd 4686 . . . . . . . 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)))
25 ssel2 3822 . . . . . . . . . 10 ((𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∧ 𝑎𝑡) → 𝑎 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠})
26 pweq 4383 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑎 → 𝒫 𝑠 = 𝒫 𝑎)
2726ineq1d 4042 . . . . . . . . . . . . . . 15 (𝑠 = 𝑎 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑎 ∩ Fin))
2827imaeq2d 5711 . . . . . . . . . . . . . 14 (𝑠 = 𝑎 → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑎 ∩ Fin)))
2928unieqd 4670 . . . . . . . . . . . . 13 (𝑠 = 𝑎 (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑎 ∩ Fin)))
30 id 22 . . . . . . . . . . . . 13 (𝑠 = 𝑎𝑠 = 𝑎)
3129, 30sseq12d 3859 . . . . . . . . . . . 12 (𝑠 = 𝑎 → ( (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠 (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎))
3231elrab 3585 . . . . . . . . . . 11 (𝑎 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝑎 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎))
3332simprbi 492 . . . . . . . . . 10 (𝑎 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} → (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎)
3425, 33syl 17 . . . . . . . . 9 ((𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∧ 𝑎𝑡) → (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎)
3534adantll 705 . . . . . . . 8 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → (𝐹 “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎)
3624, 35sstrd 3837 . . . . . . 7 ((((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) ∧ 𝑎𝑡) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑎)
3736ralrimiva 3175 . . . . . 6 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → ∀𝑎𝑡 (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑎)
38 ssint 4715 . . . . . 6 ( (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑡 ↔ ∀𝑎𝑡 (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑎)
3937, 38sylibr 226 . . . . 5 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ 𝑡)
4014, 39ssind 4063 . . . 4 (((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) ∧ 𝑡 ⊆ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠}) → (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ (𝑋 𝑡))
41 pweq 4383 . . . . . . . . 9 (𝑠 = (𝑋 𝑡) → 𝒫 𝑠 = 𝒫 (𝑋 𝑡))
4241ineq1d 4042 . . . . . . . 8 (𝑠 = (𝑋 𝑡) → (𝒫 𝑠 ∩ Fin) = (𝒫 (𝑋 𝑡) ∩ Fin))
4342imaeq2d 5711 . . . . . . 7 (𝑠 = (𝑋 𝑡) → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)))
4443unieqd 4670 . . . . . 6 (𝑠 = (𝑋 𝑡) → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)))
45 id 22 . . . . . 6 (𝑠 = (𝑋 𝑡) → 𝑠 = (𝑋 𝑡))
4644, 45sseq12d 3859 . . . . 5 (𝑠 = (𝑋 𝑡) → ( (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠 (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ (𝑋 𝑡)))
4746elrab 3585 . . . 4 ((𝑋 𝑡) ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ ((𝑋 𝑡) ∈ 𝒫 𝑋 (𝐹 “ (𝒫 (𝑋 𝑡) ∩ Fin)) ⊆ (𝑋 𝑡)))
486, 40, 47sylanbrc 578 . . 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})
492, 48ismred2 16623 . 2 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ∈ (Moore‘𝑋))
50 fssxp 6301 . . . 4 (𝐹:𝒫 𝑋⟶𝒫 𝑋𝐹 ⊆ (𝒫 𝑋 × 𝒫 𝑋))
51 pwexg 5080 . . . . 5 (𝑋𝑉 → 𝒫 𝑋 ∈ V)
5251, 51xpexd 7226 . . . 4 (𝑋𝑉 → (𝒫 𝑋 × 𝒫 𝑋) ∈ V)
53 ssexg 5031 . . . 4 ((𝐹 ⊆ (𝒫 𝑋 × 𝒫 𝑋) ∧ (𝒫 𝑋 × 𝒫 𝑋) ∈ V) → 𝐹 ∈ V)
5450, 52, 53syl2anr 590 . . 3 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → 𝐹 ∈ V)
55 simpr 479 . . . 4 ((𝑋𝑉𝐹:𝒫 𝑋⟶𝒫 𝑋) → 𝐹:𝒫 𝑋⟶𝒫 𝑋)
56 pweq 4383 . . . . . . . . . 10 (𝑠 = 𝑡 → 𝒫 𝑠 = 𝒫 𝑡)
5756ineq1d 4042 . . . . . . . . 9 (𝑠 = 𝑡 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑡 ∩ Fin))
5857imaeq2d 5711 . . . . . . . 8 (𝑠 = 𝑡 → (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑡 ∩ Fin)))
5958unieqd 4670 . . . . . . 7 (𝑠 = 𝑡 (𝐹 “ (𝒫 𝑠 ∩ Fin)) = (𝐹 “ (𝒫 𝑡 ∩ Fin)))
60 id 22 . . . . . . 7 (𝑠 = 𝑡𝑠 = 𝑡)
6159, 60sseq12d 3859 . . . . . 6 (𝑠 = 𝑡 → ( (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠 (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
6261elrab3 3587 . . . . 5 (𝑡 ∈ 𝒫 𝑋 → (𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
6362rgen 3131 . . . 4 𝑡 ∈ 𝒫 𝑋(𝑡 ∈ {𝑠 ∈ 𝒫 𝑋 (𝐹 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠} ↔ (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)
6455, 63jctir 516 . . 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)))
65 feq1 6263 . . . . 5 (𝑓 = 𝐹 → (𝑓:𝒫 𝑋⟶𝒫 𝑋𝐹:𝒫 𝑋⟶𝒫 𝑋))
66 imaeq1 5706 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 “ (𝒫 𝑡 ∩ Fin)) = (𝐹 “ (𝒫 𝑡 ∩ Fin)))
6766unieqd 4670 . . . . . . . 8 (𝑓 = 𝐹 (𝑓 “ (𝒫 𝑡 ∩ Fin)) = (𝐹 “ (𝒫 𝑡 ∩ Fin)))
6867sseq1d 3857 . . . . . . 7 (𝑓 = 𝐹 → ( (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡 (𝐹 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
6968bibi2d 334 . . . . . 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)))
7069ralbidv 3195 . . . . 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)))
7165, 70anbi12d 624 . . . 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))))
7271spcegv 3511 . . 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))))
7354, 64, 72sylc 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)))
74 isacs 16671 . 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))))
7549, 73, 74sylanbrc 578 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 198  wa 386   = wceq 1656  wex 1878  wcel 2164  wral 3117  {crab 3121  Vcvv 3414  cin 3797  wss 3798  𝒫 cpw 4380   cuni 4660   cint 4699   × cxp 5344  ran crn 5347  cima 5349  wf 6123  cfv 6127  Fincfn 8228  Moorecmre 16602  ACScacs 16605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-int 4700  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-mre 16606  df-acs 16609
This theorem is referenced by:  acsfn  16679
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