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Theorem hashmap 14335
Description: The size of the set exponential of two finite sets is the exponential of their sizes. (This is the original motivation behind the notation for set exponentiation.) (Contributed by Mario Carneiro, 5-Aug-2014.) (Proof shortened by AV, 18-Jul-2022.)
Assertion
Ref Expression
hashmap ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵)))

Proof of Theorem hashmap
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 7365 . . . . . 6 (𝑥 = ∅ → (𝐴m 𝑥) = (𝐴m ∅))
21fveq2d 6846 . . . . 5 (𝑥 = ∅ → (♯‘(𝐴m 𝑥)) = (♯‘(𝐴m ∅)))
3 fveq2 6842 . . . . . 6 (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
43oveq2d 7373 . . . . 5 (𝑥 = ∅ → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘∅)))
52, 4eqeq12d 2752 . . . 4 (𝑥 = ∅ → ((♯‘(𝐴m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴m ∅)) = ((♯‘𝐴)↑(♯‘∅))))
65imbi2d 340 . . 3 (𝑥 = ∅ → ((𝐴 ∈ Fin → (♯‘(𝐴m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴m ∅)) = ((♯‘𝐴)↑(♯‘∅)))))
7 oveq2 7365 . . . . . 6 (𝑥 = 𝑦 → (𝐴m 𝑥) = (𝐴m 𝑦))
87fveq2d 6846 . . . . 5 (𝑥 = 𝑦 → (♯‘(𝐴m 𝑥)) = (♯‘(𝐴m 𝑦)))
9 fveq2 6842 . . . . . 6 (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
109oveq2d 7373 . . . . 5 (𝑥 = 𝑦 → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘𝑦)))
118, 10eqeq12d 2752 . . . 4 (𝑥 = 𝑦 → ((♯‘(𝐴m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦))))
1211imbi2d 340 . . 3 (𝑥 = 𝑦 → ((𝐴 ∈ Fin → (♯‘(𝐴m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)))))
13 oveq2 7365 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴m 𝑥) = (𝐴m (𝑦 ∪ {𝑧})))
1413fveq2d 6846 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘(𝐴m 𝑥)) = (♯‘(𝐴m (𝑦 ∪ {𝑧}))))
15 fveq2 6842 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑧})))
1615oveq2d 7373 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))
1714, 16eqeq12d 2752 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘(𝐴m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧})))))
1817imbi2d 340 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴 ∈ Fin → (♯‘(𝐴m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))))
19 oveq2 7365 . . . . . 6 (𝑥 = 𝐵 → (𝐴m 𝑥) = (𝐴m 𝐵))
2019fveq2d 6846 . . . . 5 (𝑥 = 𝐵 → (♯‘(𝐴m 𝑥)) = (♯‘(𝐴m 𝐵)))
21 fveq2 6842 . . . . . 6 (𝑥 = 𝐵 → (♯‘𝑥) = (♯‘𝐵))
2221oveq2d 7373 . . . . 5 (𝑥 = 𝐵 → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘𝐵)))
2320, 22eqeq12d 2752 . . . 4 (𝑥 = 𝐵 → ((♯‘(𝐴m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵))))
2423imbi2d 340 . . 3 (𝑥 = 𝐵 → ((𝐴 ∈ Fin → (♯‘(𝐴m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵)))))
25 hashcl 14256 . . . . . 6 (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
2625nn0cnd 12475 . . . . 5 (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℂ)
2726exp0d 14045 . . . 4 (𝐴 ∈ Fin → ((♯‘𝐴)↑0) = 1)
28 hash0 14267 . . . . . 6 (♯‘∅) = 0
2928oveq2i 7368 . . . . 5 ((♯‘𝐴)↑(♯‘∅)) = ((♯‘𝐴)↑0)
3029a1i 11 . . . 4 (𝐴 ∈ Fin → ((♯‘𝐴)↑(♯‘∅)) = ((♯‘𝐴)↑0))
31 mapdm0 8780 . . . . . 6 (𝐴 ∈ Fin → (𝐴m ∅) = {∅})
3231fveq2d 6846 . . . . 5 (𝐴 ∈ Fin → (♯‘(𝐴m ∅)) = (♯‘{∅}))
33 0ex 5264 . . . . . 6 ∅ ∈ V
34 hashsng 14269 . . . . . 6 (∅ ∈ V → (♯‘{∅}) = 1)
3533, 34mp1i 13 . . . . 5 (𝐴 ∈ Fin → (♯‘{∅}) = 1)
3632, 35eqtrd 2776 . . . 4 (𝐴 ∈ Fin → (♯‘(𝐴m ∅)) = 1)
3727, 30, 363eqtr4rd 2787 . . 3 (𝐴 ∈ Fin → (♯‘(𝐴m ∅)) = ((♯‘𝐴)↑(♯‘∅)))
38 oveq1 7364 . . . . . 6 ((♯‘(𝐴m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → ((♯‘(𝐴m 𝑦)) · (♯‘𝐴)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
39 vex 3449 . . . . . . . . . . 11 𝑦 ∈ V
4039a1i 11 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑦 ∈ V)
41 vsnex 5386 . . . . . . . . . . 11 {𝑧} ∈ V
4241a1i 11 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → {𝑧} ∈ V)
43 elex 3463 . . . . . . . . . . 11 (𝐴 ∈ Fin → 𝐴 ∈ V)
4443adantr 481 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝐴 ∈ V)
45 simprr 771 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ¬ 𝑧𝑦)
46 disjsn 4672 . . . . . . . . . . 11 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
4745, 46sylibr 233 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
48 mapunen 9090 . . . . . . . . . 10 (((𝑦 ∈ V ∧ {𝑧} ∈ V ∧ 𝐴 ∈ V) ∧ (𝑦 ∩ {𝑧}) = ∅) → (𝐴m (𝑦 ∪ {𝑧})) ≈ ((𝐴m 𝑦) × (𝐴m {𝑧})))
4940, 42, 44, 47, 48syl31anc 1373 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐴m (𝑦 ∪ {𝑧})) ≈ ((𝐴m 𝑦) × (𝐴m {𝑧})))
50 simpl 483 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝐴 ∈ Fin)
51 simprl 769 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑦 ∈ Fin)
52 snfi 8988 . . . . . . . . . . . 12 {𝑧} ∈ Fin
53 unfi 9116 . . . . . . . . . . . 12 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
5451, 52, 53sylancl 586 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝑦 ∪ {𝑧}) ∈ Fin)
55 mapfi 9292 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (𝐴m (𝑦 ∪ {𝑧})) ∈ Fin)
5650, 54, 55syl2anc 584 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐴m (𝑦 ∪ {𝑧})) ∈ Fin)
57 mapfi 9292 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝐴m 𝑦) ∈ Fin)
5857adantrr 715 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐴m 𝑦) ∈ Fin)
59 mapfi 9292 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴m {𝑧}) ∈ Fin)
6050, 52, 59sylancl 586 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐴m {𝑧}) ∈ Fin)
61 xpfi 9261 . . . . . . . . . . 11 (((𝐴m 𝑦) ∈ Fin ∧ (𝐴m {𝑧}) ∈ Fin) → ((𝐴m 𝑦) × (𝐴m {𝑧})) ∈ Fin)
6258, 60, 61syl2anc 584 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((𝐴m 𝑦) × (𝐴m {𝑧})) ∈ Fin)
63 hashen 14247 . . . . . . . . . 10 (((𝐴m (𝑦 ∪ {𝑧})) ∈ Fin ∧ ((𝐴m 𝑦) × (𝐴m {𝑧})) ∈ Fin) → ((♯‘(𝐴m (𝑦 ∪ {𝑧}))) = (♯‘((𝐴m 𝑦) × (𝐴m {𝑧}))) ↔ (𝐴m (𝑦 ∪ {𝑧})) ≈ ((𝐴m 𝑦) × (𝐴m {𝑧}))))
6456, 62, 63syl2anc 584 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((♯‘(𝐴m (𝑦 ∪ {𝑧}))) = (♯‘((𝐴m 𝑦) × (𝐴m {𝑧}))) ↔ (𝐴m (𝑦 ∪ {𝑧})) ≈ ((𝐴m 𝑦) × (𝐴m {𝑧}))))
6549, 64mpbird 256 . . . . . . . 8 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (♯‘(𝐴m (𝑦 ∪ {𝑧}))) = (♯‘((𝐴m 𝑦) × (𝐴m {𝑧}))))
66 hashxp 14334 . . . . . . . . 9 (((𝐴m 𝑦) ∈ Fin ∧ (𝐴m {𝑧}) ∈ Fin) → (♯‘((𝐴m 𝑦) × (𝐴m {𝑧}))) = ((♯‘(𝐴m 𝑦)) · (♯‘(𝐴m {𝑧}))))
6758, 60, 66syl2anc 584 . . . . . . . 8 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (♯‘((𝐴m 𝑦) × (𝐴m {𝑧}))) = ((♯‘(𝐴m 𝑦)) · (♯‘(𝐴m {𝑧}))))
68 vex 3449 . . . . . . . . . . . 12 𝑧 ∈ V
6968a1i 11 . . . . . . . . . . 11 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → 𝑧 ∈ V)
7050, 69mapsnend 8980 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (𝐴m {𝑧}) ≈ 𝐴)
71 hashen 14247 . . . . . . . . . . 11 (((𝐴m {𝑧}) ∈ Fin ∧ 𝐴 ∈ Fin) → ((♯‘(𝐴m {𝑧})) = (♯‘𝐴) ↔ (𝐴m {𝑧}) ≈ 𝐴))
7260, 50, 71syl2anc 584 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((♯‘(𝐴m {𝑧})) = (♯‘𝐴) ↔ (𝐴m {𝑧}) ≈ 𝐴))
7370, 72mpbird 256 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (♯‘(𝐴m {𝑧})) = (♯‘𝐴))
7473oveq2d 7373 . . . . . . . 8 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((♯‘(𝐴m 𝑦)) · (♯‘(𝐴m {𝑧}))) = ((♯‘(𝐴m 𝑦)) · (♯‘𝐴)))
7565, 67, 743eqtrd 2780 . . . . . . 7 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (♯‘(𝐴m (𝑦 ∪ {𝑧}))) = ((♯‘(𝐴m 𝑦)) · (♯‘𝐴)))
76 hashunsng 14292 . . . . . . . . . . 11 (𝑧 ∈ V → ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1)))
7776elv 3451 . . . . . . . . . 10 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1))
7877adantl 482 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1))
7978oveq2d 7373 . . . . . . . 8 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑((♯‘𝑦) + 1)))
8026adantr 481 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (♯‘𝐴) ∈ ℂ)
81 hashcl 14256 . . . . . . . . . 10 (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ0)
8281ad2antrl 726 . . . . . . . . 9 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → (♯‘𝑦) ∈ ℕ0)
8380, 82expp1d 14052 . . . . . . . 8 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((♯‘𝐴)↑((♯‘𝑦) + 1)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
8479, 83eqtrd 2776 . . . . . . 7 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
8575, 84eqeq12d 2752 . . . . . 6 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((♯‘(𝐴m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) ↔ ((♯‘(𝐴m 𝑦)) · (♯‘𝐴)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴))))
8638, 85syl5ibr 245 . . . . 5 ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧𝑦)) → ((♯‘(𝐴m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → (♯‘(𝐴m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧})))))
8786expcom 414 . . . 4 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → (𝐴 ∈ Fin → ((♯‘(𝐴m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → (♯‘(𝐴m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))))
8887a2d 29 . . 3 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((𝐴 ∈ Fin → (♯‘(𝐴m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦))) → (𝐴 ∈ Fin → (♯‘(𝐴m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))))
896, 12, 18, 24, 37, 88findcard2s 9109 . 2 (𝐵 ∈ Fin → (𝐴 ∈ Fin → (♯‘(𝐴m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵))))
9089impcom 408 1 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  cun 3908  cin 3909  c0 4282  {csn 4586   class class class wbr 5105   × cxp 5631  cfv 6496  (class class class)co 7357  m cmap 8765  cen 8880  Fincfn 8883  cc 11049  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  0cn0 12413  cexp 13967  chash 14230
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  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  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-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  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-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  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-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-seq 13907  df-exp 13968  df-hash 14231
This theorem is referenced by:  hashpw  14336  hashwrdn  14435  prmreclem2  16789  efmndhash  18686  birthdaylem2  26302
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