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Theorem acsfn 17539
Description: Algebraicity of a conditional point closure condition. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
acsfn (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇𝑎𝐾𝑎)} ∈ (ACS‘𝑋))
Distinct variable groups:   𝐾,𝑎   𝑇,𝑎   𝑉,𝑎   𝑋,𝑎

Proof of Theorem acsfn
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funmpt 6539 . . . . . . 7 Fun (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))
2 funiunfv 7195 . . . . . . 7 (Fun (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) → 𝑐 ∈ (𝒫 𝑎 ∩ Fin)((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)))
31, 2mp1i 13 . . . . . 6 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑐 ∈ (𝒫 𝑎 ∩ Fin)((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)))
4 elinel1 4155 . . . . . . . . . . . 12 (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝑐 ∈ 𝒫 𝑎)
54elpwid 4569 . . . . . . . . . . 11 (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝑐𝑎)
6 elpwi 4567 . . . . . . . . . . 11 (𝑎 ∈ 𝒫 𝑋𝑎𝑋)
75, 6sylan9ssr 3958 . . . . . . . . . 10 ((𝑎 ∈ 𝒫 𝑋𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐𝑋)
8 velpw 4565 . . . . . . . . . 10 (𝑐 ∈ 𝒫 𝑋𝑐𝑋)
97, 8sylibr 233 . . . . . . . . 9 ((𝑎 ∈ 𝒫 𝑋𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ∈ 𝒫 𝑋)
109adantll 712 . . . . . . . 8 (((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ∈ 𝒫 𝑋)
11 eqeq1 2740 . . . . . . . . . 10 (𝑏 = 𝑐 → (𝑏 = 𝑇𝑐 = 𝑇))
1211ifbid 4509 . . . . . . . . 9 (𝑏 = 𝑐 → if(𝑏 = 𝑇, {𝐾}, ∅) = if(𝑐 = 𝑇, {𝐾}, ∅))
13 eqid 2736 . . . . . . . . 9 (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) = (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))
14 snex 5388 . . . . . . . . . 10 {𝐾} ∈ V
15 0ex 5264 . . . . . . . . . 10 ∅ ∈ V
1614, 15ifex 4536 . . . . . . . . 9 if(𝑐 = 𝑇, {𝐾}, ∅) ∈ V
1712, 13, 16fvmpt 6948 . . . . . . . 8 (𝑐 ∈ 𝒫 𝑋 → ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = if(𝑐 = 𝑇, {𝐾}, ∅))
1810, 17syl 17 . . . . . . 7 (((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = if(𝑐 = 𝑇, {𝐾}, ∅))
1918iuneq2dv 4978 . . . . . 6 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑐 ∈ (𝒫 𝑎 ∩ Fin)((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅))
203, 19eqtr3d 2778 . . . . 5 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) = 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅))
2120sseq1d 3975 . . . 4 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ( ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎))
22 iunss 5005 . . . . 5 ( 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎)
23 sseq1 3969 . . . . . . . . 9 ({𝐾} = if(𝑐 = 𝑇, {𝐾}, ∅) → ({𝐾} ⊆ 𝑎 ↔ if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎))
2423bibi1d 343 . . . . . . . 8 ({𝐾} = if(𝑐 = 𝑇, {𝐾}, ∅) → (({𝐾} ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)) ↔ (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎))))
25 sseq1 3969 . . . . . . . . 9 (∅ = if(𝑐 = 𝑇, {𝐾}, ∅) → (∅ ⊆ 𝑎 ↔ if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎))
2625bibi1d 343 . . . . . . . 8 (∅ = if(𝑐 = 𝑇, {𝐾}, ∅) → ((∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)) ↔ (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎))))
27 snssg 4744 . . . . . . . . . 10 (𝐾𝑋 → (𝐾𝑎 ↔ {𝐾} ⊆ 𝑎))
2827adantr 481 . . . . . . . . 9 ((𝐾𝑋𝑐 = 𝑇) → (𝐾𝑎 ↔ {𝐾} ⊆ 𝑎))
29 biimt 360 . . . . . . . . . 10 (𝑐 = 𝑇 → (𝐾𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
3029adantl 482 . . . . . . . . 9 ((𝐾𝑋𝑐 = 𝑇) → (𝐾𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
3128, 30bitr3d 280 . . . . . . . 8 ((𝐾𝑋𝑐 = 𝑇) → ({𝐾} ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
32 0ss 4356 . . . . . . . . . . 11 ∅ ⊆ 𝑎
3332a1i 11 . . . . . . . . . 10 𝑐 = 𝑇 → ∅ ⊆ 𝑎)
34 pm2.21 123 . . . . . . . . . 10 𝑐 = 𝑇 → (𝑐 = 𝑇𝐾𝑎))
3533, 342thd 264 . . . . . . . . 9 𝑐 = 𝑇 → (∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
3635adantl 482 . . . . . . . 8 ((𝐾𝑋 ∧ ¬ 𝑐 = 𝑇) → (∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
3724, 26, 31, 36ifbothda 4524 . . . . . . 7 (𝐾𝑋 → (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
3837ralbidv 3174 . . . . . 6 (𝐾𝑋 → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇𝐾𝑎)))
3938ad3antlr 729 . . . . 5 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇𝐾𝑎)))
4022, 39bitrid 282 . . . 4 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ( 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇𝐾𝑎)))
41 inss1 4188 . . . . . . . 8 (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑎
426sspwd 4573 . . . . . . . 8 (𝑎 ∈ 𝒫 𝑋 → 𝒫 𝑎 ⊆ 𝒫 𝑋)
4341, 42sstrid 3955 . . . . . . 7 (𝑎 ∈ 𝒫 𝑋 → (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑋)
4443adantl 482 . . . . . 6 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑋)
45 ralss 4014 . . . . . 6 ((𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑋 → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇𝐾𝑎) ↔ ∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇𝐾𝑎))))
4644, 45syl 17 . . . . 5 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇𝐾𝑎) ↔ ∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇𝐾𝑎))))
47 bi2.04 388 . . . . . . 7 ((𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇𝐾𝑎)) ↔ (𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)))
4847ralbii 3096 . . . . . 6 (∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇𝐾𝑎)) ↔ ∀𝑐 ∈ 𝒫 𝑋(𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)))
49 elpwg 4563 . . . . . . . . 9 (𝑇 ∈ Fin → (𝑇 ∈ 𝒫 𝑋𝑇𝑋))
5049biimparc 480 . . . . . . . 8 ((𝑇𝑋𝑇 ∈ Fin) → 𝑇 ∈ 𝒫 𝑋)
5150ad2antlr 725 . . . . . . 7 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑇 ∈ 𝒫 𝑋)
52 eleq1 2825 . . . . . . . . 9 (𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) ↔ 𝑇 ∈ (𝒫 𝑎 ∩ Fin)))
5352imbi1d 341 . . . . . . . 8 (𝑐 = 𝑇 → ((𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)))
5453ceqsralv 3483 . . . . . . 7 (𝑇 ∈ 𝒫 𝑋 → (∀𝑐 ∈ 𝒫 𝑋(𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)))
5551, 54syl 17 . . . . . 6 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ 𝒫 𝑋(𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)))
5648, 55bitrid 282 . . . . 5 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇𝐾𝑎)) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)))
57 simplrr 776 . . . . . . . . 9 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑇 ∈ Fin)
5857biantrud 532 . . . . . . . 8 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ 𝒫 𝑎 ↔ (𝑇 ∈ 𝒫 𝑎𝑇 ∈ Fin)))
59 elin 3926 . . . . . . . 8 (𝑇 ∈ (𝒫 𝑎 ∩ Fin) ↔ (𝑇 ∈ 𝒫 𝑎𝑇 ∈ Fin))
6058, 59bitr4di 288 . . . . . . 7 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ 𝒫 𝑎𝑇 ∈ (𝒫 𝑎 ∩ Fin)))
61 vex 3449 . . . . . . . 8 𝑎 ∈ V
6261elpw2 5302 . . . . . . 7 (𝑇 ∈ 𝒫 𝑎𝑇𝑎)
6360, 62bitr3di 285 . . . . . 6 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ (𝒫 𝑎 ∩ Fin) ↔ 𝑇𝑎))
6463imbi1d 341 . . . . 5 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎) ↔ (𝑇𝑎𝐾𝑎)))
6546, 56, 643bitrd 304 . . . 4 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇𝐾𝑎) ↔ (𝑇𝑎𝐾𝑎)))
6621, 40, 653bitrrd 305 . . 3 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑇𝑎𝐾𝑎) ↔ ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎))
6766rabbidva 3414 . 2 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇𝑎𝐾𝑎)} = {𝑎 ∈ 𝒫 𝑋 ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎})
68 simpll 765 . . 3 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → 𝑋𝑉)
69 snelpwi 5400 . . . . . . 7 (𝐾𝑋 → {𝐾} ∈ 𝒫 𝑋)
7069ad2antlr 725 . . . . . 6 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → {𝐾} ∈ 𝒫 𝑋)
71 0elpw 5311 . . . . . 6 ∅ ∈ 𝒫 𝑋
72 ifcl 4531 . . . . . 6 (({𝐾} ∈ 𝒫 𝑋 ∧ ∅ ∈ 𝒫 𝑋) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋)
7370, 71, 72sylancl 586 . . . . 5 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋)
7473adantr 481 . . . 4 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑏 ∈ 𝒫 𝑋) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋)
7574fmpttd 7063 . . 3 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)):𝒫 𝑋⟶𝒫 𝑋)
76 isacs1i 17537 . . 3 ((𝑋𝑉 ∧ (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)):𝒫 𝑋⟶𝒫 𝑋) → {𝑎 ∈ 𝒫 𝑋 ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎} ∈ (ACS‘𝑋))
7768, 75, 76syl2anc 584 . 2 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎} ∈ (ACS‘𝑋))
7867, 77eqeltrd 2838 1 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇𝑎𝐾𝑎)} ∈ (ACS‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  {crab 3407  cin 3909  wss 3910  c0 4282  ifcif 4486  𝒫 cpw 4560  {csn 4586   cuni 4865   ciun 4954  cmpt 5188  cima 5636  Fun wfun 6490  wf 6492  cfv 6496  Fincfn 8883  ACScacs 17465
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-mre 17466  df-acs 17469
This theorem is referenced by:  acsfn0  17540  acsfn1  17541  acsfn2  17543  acsfn1p  20266
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