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Theorem hashmap 14483

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) → (♯‘(𝐴 ↑_m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵)))

Proof of Theorem hashmap Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 7424	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ↑_m 𝑥) = (𝐴 ↑_m ∅))
2	1	fveq2d 6889	. . . . 5 ⊢ (𝑥 = ∅ → (♯‘(𝐴 ↑_m 𝑥)) = (♯‘(𝐴 ↑_m ∅)))
3		fveq2 6885	. . . . . 6 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
4	3	oveq2d 7432	. . . . 5 ⊢ (𝑥 = ∅ → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘∅)))
5	2, 4	eqeq12d 2782	. . . 4 ⊢ (𝑥 = ∅ → ((♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴 ↑_m ∅)) = ((♯‘𝐴)↑(♯‘∅))))
6	5	imbi2d 343	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m ∅)) = ((♯‘𝐴)↑(♯‘∅)))))
7		oveq2 7424	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ↑_m 𝑥) = (𝐴 ↑_m 𝑦))
8	7	fveq2d 6889	. . . . 5 ⊢ (𝑥 = 𝑦 → (♯‘(𝐴 ↑_m 𝑥)) = (♯‘(𝐴 ↑_m 𝑦)))
9		fveq2 6885	. . . . . 6 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
10	9	oveq2d 7432	. . . . 5 ⊢ (𝑥 = 𝑦 → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘𝑦)))
11	8, 10	eqeq12d 2782	. . . 4 ⊢ (𝑥 = 𝑦 → ((♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦))))
12	11	imbi2d 343	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)))))
13		oveq2 7424	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ↑_m 𝑥) = (𝐴 ↑_m (𝑦 ∪ {𝑧})))
14	13	fveq2d 6889	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘(𝐴 ↑_m 𝑥)) = (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))))
15		fveq2 6885	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑧})))
16	15	oveq2d 7432	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))
17	14, 16	eqeq12d 2782	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧})))))
18	17	imbi2d 343	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))))
19		oveq2 7424	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ↑_m 𝑥) = (𝐴 ↑_m 𝐵))
20	19	fveq2d 6889	. . . . 5 ⊢ (𝑥 = 𝐵 → (♯‘(𝐴 ↑_m 𝑥)) = (♯‘(𝐴 ↑_m 𝐵)))
21		fveq2 6885	. . . . . 6 ⊢ (𝑥 = 𝐵 → (♯‘𝑥) = (♯‘𝐵))
22	21	oveq2d 7432	. . . . 5 ⊢ (𝑥 = 𝐵 → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘𝐵)))
23	20, 22	eqeq12d 2782	. . . 4 ⊢ (𝑥 = 𝐵 → ((♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴 ↑_m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵))))
24	23	imbi2d 343	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵)))))
25		hashcl 14403	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
26	25	nn0cnd 12577	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℂ)
27	26	exp0d 14187	. . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴)↑0) = 1)
28		hash0 14414	. . . . . 6 ⊢ (♯‘∅) = 0
29	28	oveq2i 7427	. . . . 5 ⊢ ((♯‘𝐴)↑(♯‘∅)) = ((♯‘𝐴)↑0)
30	29	a1i 11	. . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴)↑(♯‘∅)) = ((♯‘𝐴)↑0))
31		mapdm0 8841	. . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ↑_m ∅) = {∅})
32	31	fveq2d 6889	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m ∅)) = (♯‘{∅}))
33		0ex 5273	. . . . . 6 ⊢ ∅ ∈ V
34		hashsng 14416	. . . . . 6 ⊢ (∅ ∈ V → (♯‘{∅}) = 1)
35	33, 34	mp1i 14	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘{∅}) = 1)
36	32, 35	eqtrd 2801	. . . 4 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m ∅)) = 1)
37	27, 30, 36	3eqtr4rd 2812	. . 3 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m ∅)) = ((♯‘𝐴)↑(♯‘∅)))
38		oveq1 7423	. . . . . 6 ⊢ ((♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘𝐴)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
39		vex 3462	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
40	39	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ V)
41		vsnex 5409	. . . . . . . . . . 11 ⊢ {𝑧} ∈ V
42	41	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → {𝑧} ∈ V)
43		elex 3479	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ V)
44	43	adantr 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝐴 ∈ V)
45		simprr 785	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ¬ 𝑧 ∈ 𝑦)
46		disjsn 4680	. . . . . . . . . . 11 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
47	45, 46	sylibr 237	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
48		mapunen 9136	. . . . . . . . . 10 ⊢ (((𝑦 ∈ V ∧ {𝑧} ∈ V ∧ 𝐴 ∈ V) ∧ (𝑦 ∩ {𝑧}) = ∅) → (𝐴 ↑_m (𝑦 ∪ {𝑧})) ≈ ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧})))
49	40, 42, 44, 47, 48	syl31anc 1400	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m (𝑦 ∪ {𝑧})) ≈ ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧})))
50		simpl 488	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝐴 ∈ Fin)
51		simprl 783	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ Fin)
52		snfi 9042	. . . . . . . . . . . 12 ⊢ {𝑧} ∈ Fin
53		unfi 9157	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
54	51, 52, 53	sylancl 598	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑦 ∪ {𝑧}) ∈ Fin)
55		mapfi 9307	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (𝐴 ↑_m (𝑦 ∪ {𝑧})) ∈ Fin)
56	50, 54, 55	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m (𝑦 ∪ {𝑧})) ∈ Fin)
57		mapfi 9307	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝐴 ↑_m 𝑦) ∈ Fin)
58	57	adantrr 730	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m 𝑦) ∈ Fin)
59		mapfi 9307	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ↑_m {𝑧}) ∈ Fin)
60	50, 52, 59	sylancl 598	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m {𝑧}) ∈ Fin)
61		xpfi 9281	. . . . . . . . . . 11 ⊢ (((𝐴 ↑_m 𝑦) ∈ Fin ∧ (𝐴 ↑_m {𝑧}) ∈ Fin) → ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧})) ∈ Fin)
62	58, 60, 61	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧})) ∈ Fin)
63		hashen 14394	. . . . . . . . . 10 ⊢ (((𝐴 ↑_m (𝑦 ∪ {𝑧})) ∈ Fin ∧ ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧})) ∈ Fin) → ((♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = (♯‘((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))) ↔ (𝐴 ↑_m (𝑦 ∪ {𝑧})) ≈ ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))))
64	56, 62, 63	syl2anc 596	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = (♯‘((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))) ↔ (𝐴 ↑_m (𝑦 ∪ {𝑧})) ≈ ((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))))
65	49, 64	mpbird 260	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = (♯‘((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))))
66		hashxp 14482	. . . . . . . . 9 ⊢ (((𝐴 ↑_m 𝑦) ∈ Fin ∧ (𝐴 ↑_m {𝑧}) ∈ Fin) → (♯‘((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))) = ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘(𝐴 ↑_m {𝑧}))))
67	58, 60, 66	syl2anc 596	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘((𝐴 ↑_m 𝑦) × (𝐴 ↑_m {𝑧}))) = ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘(𝐴 ↑_m {𝑧}))))
68		vex 3462	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
69	68	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑧 ∈ V)
70	50, 69	mapsnend 9035	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑_m {𝑧}) ≈ 𝐴)
71		hashen 14394	. . . . . . . . . . 11 ⊢ (((𝐴 ↑_m {𝑧}) ∈ Fin ∧ 𝐴 ∈ Fin) → ((♯‘(𝐴 ↑_m {𝑧})) = (♯‘𝐴) ↔ (𝐴 ↑_m {𝑧}) ≈ 𝐴))
72	60, 50, 71	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑_m {𝑧})) = (♯‘𝐴) ↔ (𝐴 ↑_m {𝑧}) ≈ 𝐴))
73	70, 72	mpbird 260	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝐴 ↑_m {𝑧})) = (♯‘𝐴))
74	73	oveq2d 7432	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘(𝐴 ↑_m {𝑧}))) = ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘𝐴)))
75	65, 67, 74	3eqtrd 2805	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘𝐴)))
76		hashunsng 14439	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1)))
77	76	elv 3463	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1))
78	77	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1))
79	78	oveq2d 7432	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑((♯‘𝑦) + 1)))
80	26	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘𝐴) ∈ ℂ)
81		hashcl 14403	. . . . . . . . . 10 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ₀)
82	81	ad2antrl 741	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘𝑦) ∈ ℕ₀)
83	80, 82	expp1d 14194	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘𝐴)↑((♯‘𝑦) + 1)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
84	79, 83	eqtrd 2801	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
85	75, 84	eqeq12d 2782	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) ↔ ((♯‘(𝐴 ↑_m 𝑦)) · (♯‘𝐴)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴))))
86	38, 85	imbitrrid 249	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧})))))
87	86	expcom 419	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝐴 ∈ Fin → ((♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))))
88	87	a2d 30	. . 3 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦))) → (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))))
89	6, 12, 18, 24, 37, 88	findcard2s 9152	. 2 ⊢ (𝐵 ∈ Fin → (𝐴 ∈ Fin → (♯‘(𝐴 ↑_m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵))))
90	89	impcom 413	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ↑_m 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 Vcvv 3458 ∪ cun 3906 ∩ cin 3907 ∅c0 4289 {csn 4592 class class class wbr 5112 × cxp 5662 ‘cfv 6540 (class class class)co 7416 ↑_m cmap 8826 ≈ cen 8942 Fincfn 8945 ℂcc 11108 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 ℕ₀cn0 12514 ↑cexp 14108 ♯chash 14377

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-sep 5260 ax-nul 5272 ax-pow 5339 ax-pr 5407 ax-un 7738 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4876 df-int 4916 df-iun 4961 df-br 5113 df-opab 5177 df-mpt 5196 df-tr 5222 df-id 5559 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7865 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-oadd 8459 df-er 8696 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-dju 9898 df-card 9936 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11453 df-neg 11454 df-nn 12244 df-n0 12515 df-z 12602 df-uz 12873 df-fz 13546 df-seq 14049 df-exp 14109 df-hash 14378

This theorem is used by: hashpw 14484 hashwrdn 14595 prmreclem2 16987 efmndhash 18945 birthdaylem2 27132

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