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Theorem acsfn 16526
 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 6069 . . . . . . 7 Fun (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))
2 funiunfv 6648 . . . . . . 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 inss1 3979 . . . . . . . . . . . . 13 (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑎
54sseli 3746 . . . . . . . . . . . 12 (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝑐 ∈ 𝒫 𝑎)
65elpwid 4307 . . . . . . . . . . 11 (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝑐𝑎)
7 elpwi 4305 . . . . . . . . . . 11 (𝑎 ∈ 𝒫 𝑋𝑎𝑋)
86, 7sylan9ssr 3764 . . . . . . . . . 10 ((𝑎 ∈ 𝒫 𝑋𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐𝑋)
9 selpw 4302 . . . . . . . . . 10 (𝑐 ∈ 𝒫 𝑋𝑐𝑋)
108, 9sylibr 224 . . . . . . . . 9 ((𝑎 ∈ 𝒫 𝑋𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ∈ 𝒫 𝑋)
1110adantll 685 . . . . . . . 8 (((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ∈ 𝒫 𝑋)
12 eqeq1 2774 . . . . . . . . . 10 (𝑏 = 𝑐 → (𝑏 = 𝑇𝑐 = 𝑇))
1312ifbid 4245 . . . . . . . . 9 (𝑏 = 𝑐 → if(𝑏 = 𝑇, {𝐾}, ∅) = if(𝑐 = 𝑇, {𝐾}, ∅))
14 eqid 2770 . . . . . . . . 9 (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) = (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))
15 snex 5036 . . . . . . . . . 10 {𝐾} ∈ V
16 0ex 4921 . . . . . . . . . 10 ∅ ∈ V
1715, 16ifex 4293 . . . . . . . . 9 if(𝑐 = 𝑇, {𝐾}, ∅) ∈ V
1813, 14, 17fvmpt 6424 . . . . . . . 8 (𝑐 ∈ 𝒫 𝑋 → ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = if(𝑐 = 𝑇, {𝐾}, ∅))
1911, 18syl 17 . . . . . . 7 (((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))‘𝑐) = if(𝑐 = 𝑇, {𝐾}, ∅))
2019iuneq2dv 4674 . . . . . 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}, ∅))
213, 20eqtr3d 2806 . . . . 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}, ∅))
2221sseq1d 3779 . . . 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))
23 iunss 4693 . . . . 5 ( 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎)
24 sseq1 3773 . . . . . . . . 9 ({𝐾} = if(𝑐 = 𝑇, {𝐾}, ∅) → ({𝐾} ⊆ 𝑎 ↔ if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎))
2524bibi1d 332 . . . . . . . 8 ({𝐾} = if(𝑐 = 𝑇, {𝐾}, ∅) → (({𝐾} ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)) ↔ (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎))))
26 sseq1 3773 . . . . . . . . 9 (∅ = if(𝑐 = 𝑇, {𝐾}, ∅) → (∅ ⊆ 𝑎 ↔ if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎))
2726bibi1d 332 . . . . . . . 8 (∅ = if(𝑐 = 𝑇, {𝐾}, ∅) → ((∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)) ↔ (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎))))
28 snssg 4448 . . . . . . . . . 10 (𝐾𝑋 → (𝐾𝑎 ↔ {𝐾} ⊆ 𝑎))
2928adantr 466 . . . . . . . . 9 ((𝐾𝑋𝑐 = 𝑇) → (𝐾𝑎 ↔ {𝐾} ⊆ 𝑎))
30 biimt 349 . . . . . . . . . 10 (𝑐 = 𝑇 → (𝐾𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
3130adantl 467 . . . . . . . . 9 ((𝐾𝑋𝑐 = 𝑇) → (𝐾𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
3229, 31bitr3d 270 . . . . . . . 8 ((𝐾𝑋𝑐 = 𝑇) → ({𝐾} ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
33 0ss 4114 . . . . . . . . . . 11 ∅ ⊆ 𝑎
3433a1i 11 . . . . . . . . . 10 𝑐 = 𝑇 → ∅ ⊆ 𝑎)
35 pm2.21 121 . . . . . . . . . 10 𝑐 = 𝑇 → (𝑐 = 𝑇𝐾𝑎))
3634, 352thd 255 . . . . . . . . 9 𝑐 = 𝑇 → (∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
3736adantl 467 . . . . . . . 8 ((𝐾𝑋 ∧ ¬ 𝑐 = 𝑇) → (∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
3825, 27, 32, 37ifbothda 4260 . . . . . . 7 (𝐾𝑋 → (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
3938ralbidv 3134 . . . . . 6 (𝐾𝑋 → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇𝐾𝑎)))
4039ad3antlr 702 . . . . 5 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇𝐾𝑎)))
4123, 40syl5bb 272 . . . 4 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ( 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇𝐾𝑎)))
42 sspwb 5045 . . . . . . . . 9 (𝑎𝑋 ↔ 𝒫 𝑎 ⊆ 𝒫 𝑋)
437, 42sylib 208 . . . . . . . 8 (𝑎 ∈ 𝒫 𝑋 → 𝒫 𝑎 ⊆ 𝒫 𝑋)
444, 43syl5ss 3761 . . . . . . 7 (𝑎 ∈ 𝒫 𝑋 → (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑋)
4544adantl 467 . . . . . 6 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑋)
46 ralss 3815 . . . . . 6 ((𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑋 → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇𝐾𝑎) ↔ ∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇𝐾𝑎))))
4745, 46syl 17 . . . . 5 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇𝐾𝑎) ↔ ∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇𝐾𝑎))))
48 bi2.04 375 . . . . . . 7 ((𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇𝐾𝑎)) ↔ (𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)))
4948ralbii 3128 . . . . . 6 (∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇𝐾𝑎)) ↔ ∀𝑐 ∈ 𝒫 𝑋(𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)))
50 elpwg 4303 . . . . . . . . 9 (𝑇 ∈ Fin → (𝑇 ∈ 𝒫 𝑋𝑇𝑋))
5150biimparc 465 . . . . . . . 8 ((𝑇𝑋𝑇 ∈ Fin) → 𝑇 ∈ 𝒫 𝑋)
5251ad2antlr 698 . . . . . . 7 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑇 ∈ 𝒫 𝑋)
53 eleq1 2837 . . . . . . . . 9 (𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) ↔ 𝑇 ∈ (𝒫 𝑎 ∩ Fin)))
5453imbi1d 330 . . . . . . . 8 (𝑐 = 𝑇 → ((𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)))
5554ceqsralv 3383 . . . . . . 7 (𝑇 ∈ 𝒫 𝑋 → (∀𝑐 ∈ 𝒫 𝑋(𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)))
5652, 55syl 17 . . . . . 6 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ 𝒫 𝑋(𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)))
5749, 56syl5bb 272 . . . . 5 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇𝐾𝑎)) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)))
58 vex 3352 . . . . . . . 8 𝑎 ∈ V
5958elpw2 4956 . . . . . . 7 (𝑇 ∈ 𝒫 𝑎𝑇𝑎)
60 simplrr 755 . . . . . . . . 9 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑇 ∈ Fin)
6160biantrud 515 . . . . . . . 8 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ 𝒫 𝑎 ↔ (𝑇 ∈ 𝒫 𝑎𝑇 ∈ Fin)))
62 elin 3945 . . . . . . . 8 (𝑇 ∈ (𝒫 𝑎 ∩ Fin) ↔ (𝑇 ∈ 𝒫 𝑎𝑇 ∈ Fin))
6361, 62syl6bbr 278 . . . . . . 7 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ 𝒫 𝑎𝑇 ∈ (𝒫 𝑎 ∩ Fin)))
6459, 63syl5rbbr 275 . . . . . 6 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ (𝒫 𝑎 ∩ Fin) ↔ 𝑇𝑎))
6564imbi1d 330 . . . . 5 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎) ↔ (𝑇𝑎𝐾𝑎)))
6647, 57, 653bitrd 294 . . . 4 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇𝐾𝑎) ↔ (𝑇𝑎𝐾𝑎)))
6722, 41, 663bitrrd 295 . . 3 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → ((𝑇𝑎𝐾𝑎) ↔ ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎))
6867rabbidva 3337 . 2 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇𝑎𝐾𝑎)} = {𝑎 ∈ 𝒫 𝑋 ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎})
69 simpll 742 . . 3 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → 𝑋𝑉)
70 snelpwi 5040 . . . . . . 7 (𝐾𝑋 → {𝐾} ∈ 𝒫 𝑋)
7170ad2antlr 698 . . . . . 6 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → {𝐾} ∈ 𝒫 𝑋)
72 0elpw 4962 . . . . . 6 ∅ ∈ 𝒫 𝑋
73 ifcl 4267 . . . . . 6 (({𝐾} ∈ 𝒫 𝑋 ∧ ∅ ∈ 𝒫 𝑋) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋)
7471, 72, 73sylancl 566 . . . . 5 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋)
7574adantr 466 . . . 4 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑏 ∈ 𝒫 𝑋) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋)
7675, 14fmptd 6527 . . 3 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)):𝒫 𝑋⟶𝒫 𝑋)
77 isacs1i 16524 . . 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))
7869, 76, 77syl2anc 565 . 2 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎} ∈ (ACS‘𝑋))
7968, 78eqeltrd 2849 1 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇𝑎𝐾𝑎)} ∈ (ACS‘𝑋))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1630   ∈ wcel 2144  ∀wral 3060  {crab 3064   ∩ cin 3720   ⊆ wss 3721  ∅c0 4061  ifcif 4223  𝒫 cpw 4295  {csn 4314  ∪ cuni 4572  ∪ ciun 4652   ↦ cmpt 4861   “ cima 5252  Fun wfun 6025  ⟶wf 6027  ‘cfv 6031  Fincfn 8108  ACScacs 16452 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-mre 16453  df-acs 16456 This theorem is referenced by:  acsfn0  16527  acsfn1  16528  acsfn2  16530  acsfn1p  38288
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