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Theorem hashbc 13275
Description: The binomial coefficient counts the number of subsets of a finite set of a given size. This is Metamath 100 proof #58 (formula for the number of combinations). (Contributed by Mario Carneiro, 13-Jul-2014.)
Assertion
Ref Expression
hashbc ((𝐴 ∈ Fin ∧ 𝐾 ∈ ℤ) → ((#‘𝐴)C𝐾) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾}))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐾

Proof of Theorem hashbc
Dummy variables 𝑗 𝑘 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6229 . . . . . 6 (𝑤 = ∅ → (#‘𝑤) = (#‘∅))
21oveq1d 6705 . . . . 5 (𝑤 = ∅ → ((#‘𝑤)C𝑘) = ((#‘∅)C𝑘))
3 pweq 4194 . . . . . . 7 (𝑤 = ∅ → 𝒫 𝑤 = 𝒫 ∅)
4 rabeq 3223 . . . . . . 7 (𝒫 𝑤 = 𝒫 ∅ → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})
53, 4syl 17 . . . . . 6 (𝑤 = ∅ → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})
65fveq2d 6233 . . . . 5 (𝑤 = ∅ → (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
72, 6eqeq12d 2666 . . . 4 (𝑤 = ∅ → (((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})))
87ralbidv 3015 . . 3 (𝑤 = ∅ → (∀𝑘 ∈ ℤ ((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})))
9 fveq2 6229 . . . . . 6 (𝑤 = 𝑦 → (#‘𝑤) = (#‘𝑦))
109oveq1d 6705 . . . . 5 (𝑤 = 𝑦 → ((#‘𝑤)C𝑘) = ((#‘𝑦)C𝑘))
11 pweq 4194 . . . . . . 7 (𝑤 = 𝑦 → 𝒫 𝑤 = 𝒫 𝑦)
12 rabeq 3223 . . . . . . 7 (𝒫 𝑤 = 𝒫 𝑦 → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘})
1311, 12syl 17 . . . . . 6 (𝑤 = 𝑦 → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘})
1413fveq2d 6233 . . . . 5 (𝑤 = 𝑦 → (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘}))
1510, 14eqeq12d 2666 . . . 4 (𝑤 = 𝑦 → (((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘𝑦)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘})))
1615ralbidv 3015 . . 3 (𝑤 = 𝑦 → (∀𝑘 ∈ ℤ ((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((#‘𝑦)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘})))
17 fveq2 6229 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → (#‘𝑤) = (#‘(𝑦 ∪ {𝑧})))
1817oveq1d 6705 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → ((#‘𝑤)C𝑘) = ((#‘(𝑦 ∪ {𝑧}))C𝑘))
19 pweq 4194 . . . . . . 7 (𝑤 = (𝑦 ∪ {𝑧}) → 𝒫 𝑤 = 𝒫 (𝑦 ∪ {𝑧}))
20 rabeq 3223 . . . . . . 7 (𝒫 𝑤 = 𝒫 (𝑦 ∪ {𝑧}) → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})
2119, 20syl 17 . . . . . 6 (𝑤 = (𝑦 ∪ {𝑧}) → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})
2221fveq2d 6233 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘}))
2318, 22eqeq12d 2666 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → (((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})))
2423ralbidv 3015 . . 3 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ ℤ ((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})))
25 fveq2 6229 . . . . . 6 (𝑤 = 𝐴 → (#‘𝑤) = (#‘𝐴))
2625oveq1d 6705 . . . . 5 (𝑤 = 𝐴 → ((#‘𝑤)C𝑘) = ((#‘𝐴)C𝑘))
27 pweq 4194 . . . . . . 7 (𝑤 = 𝐴 → 𝒫 𝑤 = 𝒫 𝐴)
28 rabeq 3223 . . . . . . 7 (𝒫 𝑤 = 𝒫 𝐴 → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘})
2927, 28syl 17 . . . . . 6 (𝑤 = 𝐴 → {𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘})
3029fveq2d 6233 . . . . 5 (𝑤 = 𝐴 → (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘}))
3126, 30eqeq12d 2666 . . . 4 (𝑤 = 𝐴 → (((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘𝐴)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘})))
3231ralbidv 3015 . . 3 (𝑤 = 𝐴 → (∀𝑘 ∈ ℤ ((#‘𝑤)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑤 ∣ (#‘𝑥) = 𝑘}) ↔ ∀𝑘 ∈ ℤ ((#‘𝐴)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘})))
33 hash0 13196 . . . . . . . . . 10 (#‘∅) = 0
3433a1i 11 . . . . . . . . 9 (𝑘 ∈ (0...0) → (#‘∅) = 0)
35 elfz1eq 12390 . . . . . . . . 9 (𝑘 ∈ (0...0) → 𝑘 = 0)
3634, 35oveq12d 6708 . . . . . . . 8 (𝑘 ∈ (0...0) → ((#‘∅)C𝑘) = (0C0))
37 0nn0 11345 . . . . . . . . 9 0 ∈ ℕ0
38 bcn0 13137 . . . . . . . . 9 (0 ∈ ℕ0 → (0C0) = 1)
3937, 38ax-mp 5 . . . . . . . 8 (0C0) = 1
4036, 39syl6eq 2701 . . . . . . 7 (𝑘 ∈ (0...0) → ((#‘∅)C𝑘) = 1)
41 pw0 4375 . . . . . . . . . 10 𝒫 ∅ = {∅}
4235eqcomd 2657 . . . . . . . . . . . 12 (𝑘 ∈ (0...0) → 0 = 𝑘)
4341raleqi 3172 . . . . . . . . . . . . 13 (∀𝑥 ∈ 𝒫 ∅(#‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} (#‘𝑥) = 𝑘)
44 0ex 4823 . . . . . . . . . . . . . 14 ∅ ∈ V
45 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑥 = ∅ → (#‘𝑥) = (#‘∅))
4645, 33syl6eq 2701 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → (#‘𝑥) = 0)
4746eqeq1d 2653 . . . . . . . . . . . . . 14 (𝑥 = ∅ → ((#‘𝑥) = 𝑘 ↔ 0 = 𝑘))
4844, 47ralsn 4254 . . . . . . . . . . . . 13 (∀𝑥 ∈ {∅} (#‘𝑥) = 𝑘 ↔ 0 = 𝑘)
4943, 48bitri 264 . . . . . . . . . . . 12 (∀𝑥 ∈ 𝒫 ∅(#‘𝑥) = 𝑘 ↔ 0 = 𝑘)
5042, 49sylibr 224 . . . . . . . . . . 11 (𝑘 ∈ (0...0) → ∀𝑥 ∈ 𝒫 ∅(#‘𝑥) = 𝑘)
51 rabid2 3148 . . . . . . . . . . 11 (𝒫 ∅ = {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘} ↔ ∀𝑥 ∈ 𝒫 ∅(#‘𝑥) = 𝑘)
5250, 51sylibr 224 . . . . . . . . . 10 (𝑘 ∈ (0...0) → 𝒫 ∅ = {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})
5341, 52syl5reqr 2700 . . . . . . . . 9 (𝑘 ∈ (0...0) → {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘} = {∅})
5453fveq2d 6233 . . . . . . . 8 (𝑘 ∈ (0...0) → (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}) = (#‘{∅}))
55 hashsng 13197 . . . . . . . . 9 (∅ ∈ V → (#‘{∅}) = 1)
5644, 55ax-mp 5 . . . . . . . 8 (#‘{∅}) = 1
5754, 56syl6eq 2701 . . . . . . 7 (𝑘 ∈ (0...0) → (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}) = 1)
5840, 57eqtr4d 2688 . . . . . 6 (𝑘 ∈ (0...0) → ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
5958adantl 481 . . . . 5 ((𝑘 ∈ ℤ ∧ 𝑘 ∈ (0...0)) → ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
6033oveq1i 6700 . . . . . 6 ((#‘∅)C𝑘) = (0C𝑘)
61 bcval3 13133 . . . . . . . 8 ((0 ∈ ℕ0𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
6237, 61mp3an1 1451 . . . . . . 7 ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = 0)
63 id 22 . . . . . . . . . . . . . 14 (0 = 𝑘 → 0 = 𝑘)
64 0z 11426 . . . . . . . . . . . . . . 15 0 ∈ ℤ
65 elfz3 12389 . . . . . . . . . . . . . . 15 (0 ∈ ℤ → 0 ∈ (0...0))
6664, 65ax-mp 5 . . . . . . . . . . . . . 14 0 ∈ (0...0)
6763, 66syl6eqelr 2739 . . . . . . . . . . . . 13 (0 = 𝑘𝑘 ∈ (0...0))
6867con3i 150 . . . . . . . . . . . 12 𝑘 ∈ (0...0) → ¬ 0 = 𝑘)
6968adantl 481 . . . . . . . . . . 11 ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ¬ 0 = 𝑘)
7041raleqi 3172 . . . . . . . . . . . 12 (∀𝑥 ∈ 𝒫 ∅ ¬ (#‘𝑥) = 𝑘 ↔ ∀𝑥 ∈ {∅} ¬ (#‘𝑥) = 𝑘)
7147notbid 307 . . . . . . . . . . . . 13 (𝑥 = ∅ → (¬ (#‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘))
7244, 71ralsn 4254 . . . . . . . . . . . 12 (∀𝑥 ∈ {∅} ¬ (#‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘)
7370, 72bitri 264 . . . . . . . . . . 11 (∀𝑥 ∈ 𝒫 ∅ ¬ (#‘𝑥) = 𝑘 ↔ ¬ 0 = 𝑘)
7469, 73sylibr 224 . . . . . . . . . 10 ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ∀𝑥 ∈ 𝒫 ∅ ¬ (#‘𝑥) = 𝑘)
75 rabeq0 3990 . . . . . . . . . 10 ({𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘} = ∅ ↔ ∀𝑥 ∈ 𝒫 ∅ ¬ (#‘𝑥) = 𝑘)
7674, 75sylibr 224 . . . . . . . . 9 ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → {𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘} = ∅)
7776fveq2d 6233 . . . . . . . 8 ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}) = (#‘∅))
7877, 33syl6eq 2701 . . . . . . 7 ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}) = 0)
7962, 78eqtr4d 2688 . . . . . 6 ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → (0C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
8060, 79syl5eq 2697 . . . . 5 ((𝑘 ∈ ℤ ∧ ¬ 𝑘 ∈ (0...0)) → ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
8159, 80pm2.61dan 849 . . . 4 (𝑘 ∈ ℤ → ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘}))
8281rgen 2951 . . 3 𝑘 ∈ ℤ ((#‘∅)C𝑘) = (#‘{𝑥 ∈ 𝒫 ∅ ∣ (#‘𝑥) = 𝑘})
83 oveq2 6698 . . . . . 6 (𝑘 = 𝑗 → ((#‘𝑦)C𝑘) = ((#‘𝑦)C𝑗))
84 eqeq2 2662 . . . . . . . . 9 (𝑘 = 𝑗 → ((#‘𝑥) = 𝑘 ↔ (#‘𝑥) = 𝑗))
8584rabbidv 3220 . . . . . . . 8 (𝑘 = 𝑗 → {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗})
86 fveq2 6229 . . . . . . . . . 10 (𝑥 = 𝑧 → (#‘𝑥) = (#‘𝑧))
8786eqeq1d 2653 . . . . . . . . 9 (𝑥 = 𝑧 → ((#‘𝑥) = 𝑗 ↔ (#‘𝑧) = 𝑗))
8887cbvrabv 3230 . . . . . . . 8 {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗} = {𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}
8985, 88syl6eq 2701 . . . . . . 7 (𝑘 = 𝑗 → {𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘} = {𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗})
9089fveq2d 6233 . . . . . 6 (𝑘 = 𝑗 → (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))
9183, 90eqeq12d 2666 . . . . 5 (𝑘 = 𝑗 → (((#‘𝑦)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗})))
9291cbvralv 3201 . . . 4 (∀𝑘 ∈ ℤ ((#‘𝑦)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘}) ↔ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))
93 simpll 805 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → 𝑦 ∈ Fin)
94 simplr 807 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → ¬ 𝑧𝑦)
95 simprr 811 . . . . . . . 8 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))
9688fveq2i 6232 . . . . . . . . . 10 (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗}) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗})
9796eqeq2i 2663 . . . . . . . . 9 (((#‘𝑦)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗}) ↔ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))
9897ralbii 3009 . . . . . . . 8 (∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗}) ↔ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))
9995, 98sylibr 224 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑗}))
100 simprl 809 . . . . . . 7 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → 𝑘 ∈ ℤ)
10193, 94, 99, 100hashbclem 13274 . . . . . 6 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ (𝑘 ∈ ℤ ∧ ∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}))) → ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘}))
102101expr 642 . . . . 5 (((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) ∧ 𝑘 ∈ ℤ) → (∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}) → ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})))
103102ralrimdva 2998 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑗 ∈ ℤ ((#‘𝑦)C𝑗) = (#‘{𝑧 ∈ 𝒫 𝑦 ∣ (#‘𝑧) = 𝑗}) → ∀𝑘 ∈ ℤ ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})))
10492, 103syl5bi 232 . . 3 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (∀𝑘 ∈ ℤ ((#‘𝑦)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝑦 ∣ (#‘𝑥) = 𝑘}) → ∀𝑘 ∈ ℤ ((#‘(𝑦 ∪ {𝑧}))C𝑘) = (#‘{𝑥 ∈ 𝒫 (𝑦 ∪ {𝑧}) ∣ (#‘𝑥) = 𝑘})))
1058, 16, 24, 32, 82, 104findcard2s 8242 . 2 (𝐴 ∈ Fin → ∀𝑘 ∈ ℤ ((#‘𝐴)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘}))
106 oveq2 6698 . . . 4 (𝑘 = 𝐾 → ((#‘𝐴)C𝑘) = ((#‘𝐴)C𝐾))
107 eqeq2 2662 . . . . . 6 (𝑘 = 𝐾 → ((#‘𝑥) = 𝑘 ↔ (#‘𝑥) = 𝐾))
108107rabbidv 3220 . . . . 5 (𝑘 = 𝐾 → {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾})
109108fveq2d 6233 . . . 4 (𝑘 = 𝐾 → (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘}) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾}))
110106, 109eqeq12d 2666 . . 3 (𝑘 = 𝐾 → (((#‘𝐴)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘}) ↔ ((#‘𝐴)C𝐾) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾})))
111110rspccva 3339 . 2 ((∀𝑘 ∈ ℤ ((#‘𝐴)C𝑘) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑘}) ∧ 𝐾 ∈ ℤ) → ((#‘𝐴)C𝐾) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾}))
112105, 111sylan 487 1 ((𝐴 ∈ Fin ∧ 𝐾 ∈ ℤ) → ((#‘𝐴)C𝐾) = (#‘{𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝐾}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  cun 3605  c0 3948  𝒫 cpw 4191  {csn 4210  cfv 5926  (class class class)co 6690  Fincfn 7997  0cc0 9974  1c1 9975  0cn0 11330  cz 11415  ...cfz 12364  Ccbc 13129  #chash 13157
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-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  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-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  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-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-seq 12842  df-fac 13101  df-bc 13130  df-hash 13158
This theorem is referenced by:  hashbc2  15757  sylow1lem1  18059  musum  24962  ballotlem1  30676  ballotlem2  30678
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