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Theorem acsfn 16314
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 5924 . . . . . . 7 Fun (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))
2 funiunfv 6503 . . . . . . 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 3831 . . . . . . . . . . . . 13 (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑎
54sseli 3597 . . . . . . . . . . . 12 (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝑐 ∈ 𝒫 𝑎)
65elpwid 4168 . . . . . . . . . . 11 (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝑐𝑎)
7 elpwi 4166 . . . . . . . . . . 11 (𝑎 ∈ 𝒫 𝑋𝑎𝑋)
86, 7sylan9ssr 3615 . . . . . . . . . 10 ((𝑎 ∈ 𝒫 𝑋𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐𝑋)
9 selpw 4163 . . . . . . . . . 10 (𝑐 ∈ 𝒫 𝑋𝑐𝑋)
108, 9sylibr 224 . . . . . . . . 9 ((𝑎 ∈ 𝒫 𝑋𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ∈ 𝒫 𝑋)
1110adantll 750 . . . . . . . 8 (((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) ∧ 𝑐 ∈ (𝒫 𝑎 ∩ Fin)) → 𝑐 ∈ 𝒫 𝑋)
12 eqeq1 2625 . . . . . . . . . 10 (𝑏 = 𝑐 → (𝑏 = 𝑇𝑐 = 𝑇))
1312ifbid 4106 . . . . . . . . 9 (𝑏 = 𝑐 → if(𝑏 = 𝑇, {𝐾}, ∅) = if(𝑐 = 𝑇, {𝐾}, ∅))
14 eqid 2621 . . . . . . . . 9 (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) = (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅))
15 snex 4906 . . . . . . . . . 10 {𝐾} ∈ V
16 0ex 4788 . . . . . . . . . 10 ∅ ∈ V
1715, 16ifex 4154 . . . . . . . . 9 if(𝑐 = 𝑇, {𝐾}, ∅) ∈ V
1813, 14, 17fvmpt 6280 . . . . . . . 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 4540 . . . . . 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 2657 . . . . 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 3630 . . . 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 4559 . . . . 5 ( 𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎)
24 sseq1 3624 . . . . . . . . 9 ({𝐾} = if(𝑐 = 𝑇, {𝐾}, ∅) → ({𝐾} ⊆ 𝑎 ↔ if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎))
2524bibi1d 333 . . . . . . . 8 ({𝐾} = if(𝑐 = 𝑇, {𝐾}, ∅) → (({𝐾} ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)) ↔ (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎))))
26 sseq1 3624 . . . . . . . . 9 (∅ = if(𝑐 = 𝑇, {𝐾}, ∅) → (∅ ⊆ 𝑎 ↔ if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎))
2726bibi1d 333 . . . . . . . 8 (∅ = if(𝑐 = 𝑇, {𝐾}, ∅) → ((∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)) ↔ (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎))))
28 snssg 4325 . . . . . . . . . 10 (𝐾𝑋 → (𝐾𝑎 ↔ {𝐾} ⊆ 𝑎))
2928adantr 481 . . . . . . . . 9 ((𝐾𝑋𝑐 = 𝑇) → (𝐾𝑎 ↔ {𝐾} ⊆ 𝑎))
30 biimt 350 . . . . . . . . . 10 (𝑐 = 𝑇 → (𝐾𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
3130adantl 482 . . . . . . . . 9 ((𝐾𝑋𝑐 = 𝑇) → (𝐾𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
3229, 31bitr3d 270 . . . . . . . 8 ((𝐾𝑋𝑐 = 𝑇) → ({𝐾} ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
33 0ss 3970 . . . . . . . . . . 11 ∅ ⊆ 𝑎
3433a1i 11 . . . . . . . . . 10 𝑐 = 𝑇 → ∅ ⊆ 𝑎)
35 pm2.21 120 . . . . . . . . . 10 𝑐 = 𝑇 → (𝑐 = 𝑇𝐾𝑎))
3634, 352thd 255 . . . . . . . . 9 𝑐 = 𝑇 → (∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
3736adantl 482 . . . . . . . 8 ((𝐾𝑋 ∧ ¬ 𝑐 = 𝑇) → (∅ ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
3825, 27, 32, 37ifbothda 4121 . . . . . . 7 (𝐾𝑋 → (if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ (𝑐 = 𝑇𝐾𝑎)))
3938ralbidv 2985 . . . . . 6 (𝐾𝑋 → (∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)if(𝑐 = 𝑇, {𝐾}, ∅) ⊆ 𝑎 ↔ ∀𝑐 ∈ (𝒫 𝑎 ∩ Fin)(𝑐 = 𝑇𝐾𝑎)))
4039ad3antlr 767 . . . . 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 4915 . . . . . . . . 9 (𝑎𝑋 ↔ 𝒫 𝑎 ⊆ 𝒫 𝑋)
437, 42sylib 208 . . . . . . . 8 (𝑎 ∈ 𝒫 𝑋 → 𝒫 𝑎 ⊆ 𝒫 𝑋)
444, 43syl5ss 3612 . . . . . . 7 (𝑎 ∈ 𝒫 𝑋 → (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑋)
4544adantl 482 . . . . . 6 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝒫 𝑎 ∩ Fin) ⊆ 𝒫 𝑋)
46 ralss 3666 . . . . . 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 bi2.04 376 . . . . . . 7 ((𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇𝐾𝑎)) ↔ (𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)))
4948ralbii 2979 . . . . . 6 (∀𝑐 ∈ 𝒫 𝑋(𝑐 ∈ (𝒫 𝑎 ∩ Fin) → (𝑐 = 𝑇𝐾𝑎)) ↔ ∀𝑐 ∈ 𝒫 𝑋(𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)))
50 elpwg 4164 . . . . . . . . 9 (𝑇 ∈ Fin → (𝑇 ∈ 𝒫 𝑋𝑇𝑋))
5150biimparc 504 . . . . . . . 8 ((𝑇𝑋𝑇 ∈ Fin) → 𝑇 ∈ 𝒫 𝑋)
5251ad2antlr 763 . . . . . . 7 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑇 ∈ 𝒫 𝑋)
53 eleq1 2688 . . . . . . . . 9 (𝑐 = 𝑇 → (𝑐 ∈ (𝒫 𝑎 ∩ Fin) ↔ 𝑇 ∈ (𝒫 𝑎 ∩ Fin)))
5453imbi1d 331 . . . . . . . 8 (𝑐 = 𝑇 → ((𝑐 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎) ↔ (𝑇 ∈ (𝒫 𝑎 ∩ Fin) → 𝐾𝑎)))
5554ceqsralv 3232 . . . . . . 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 3201 . . . . . . . 8 𝑎 ∈ V
5958elpw2 4826 . . . . . . 7 (𝑇 ∈ 𝒫 𝑎𝑇𝑎)
60 simplrr 801 . . . . . . . . 9 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → 𝑇 ∈ Fin)
6160biantrud 528 . . . . . . . 8 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑎 ∈ 𝒫 𝑋) → (𝑇 ∈ 𝒫 𝑎 ↔ (𝑇 ∈ 𝒫 𝑎𝑇 ∈ Fin)))
62 elin 3794 . . . . . . . 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 331 . . . . 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 3186 . 2 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ∣ (𝑇𝑎𝐾𝑎)} = {𝑎 ∈ 𝒫 𝑋 ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎})
69 simpll 790 . . 3 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → 𝑋𝑉)
70 snelpwi 4910 . . . . . . 7 (𝐾𝑋 → {𝐾} ∈ 𝒫 𝑋)
7170ad2antlr 763 . . . . . 6 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → {𝐾} ∈ 𝒫 𝑋)
72 0elpw 4832 . . . . . 6 ∅ ∈ 𝒫 𝑋
73 ifcl 4128 . . . . . 6 (({𝐾} ∈ 𝒫 𝑋 ∧ ∅ ∈ 𝒫 𝑋) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋)
7471, 72, 73sylancl 694 . . . . 5 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋)
7574adantr 481 . . . 4 ((((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) ∧ 𝑏 ∈ 𝒫 𝑋) → if(𝑏 = 𝑇, {𝐾}, ∅) ∈ 𝒫 𝑋)
7675, 14fmptd 6383 . . 3 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → (𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)):𝒫 𝑋⟶𝒫 𝑋)
77 isacs1i 16312 . . 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 693 . 2 (((𝑋𝑉𝐾𝑋) ∧ (𝑇𝑋𝑇 ∈ Fin)) → {𝑎 ∈ 𝒫 𝑋 ((𝑏 ∈ 𝒫 𝑋 ↦ if(𝑏 = 𝑇, {𝐾}, ∅)) “ (𝒫 𝑎 ∩ Fin)) ⊆ 𝑎} ∈ (ACS‘𝑋))
7968, 78eqeltrd 2700 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 384   = wceq 1482  wcel 1989  wral 2911  {crab 2915  cin 3571  wss 3572  c0 3913  ifcif 4084  𝒫 cpw 4156  {csn 4175   cuni 4434   ciun 4518  cmpt 4727  cima 5115  Fun wfun 5880  wf 5882  cfv 5886  Fincfn 7952  ACScacs 16239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-mre 16240  df-acs 16243
This theorem is referenced by:  acsfn0  16315  acsfn1  16316  acsfn2  16318  acsfn1p  37595
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