Theorem hashmap 11222

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 6068	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑚 𝑥) = (𝐴 ↑𝑚 ∅))
2	1	fveq2d 5681	. . . . 5 ⊢ (𝑥 = ∅ → (♯‘(𝐴 ↑𝑚 𝑥)) = (♯‘(𝐴 ↑𝑚 ∅)))
3		fveq2 5677	. . . . . 6 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
4	3	oveq2d 6076	. . . . 5 ⊢ (𝑥 = ∅ → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘∅)))
5	2, 4	eqeq12d 2249	. . . 4 ⊢ (𝑥 = ∅ → ((♯‘(𝐴 ↑𝑚 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴 ↑𝑚 ∅)) = ((♯‘𝐴)↑(♯‘∅))))
6	5	imbi2d 230	. . 3 ⊢ (𝑥 = ∅ → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 ∅)) = ((♯‘𝐴)↑(♯‘∅)))))
7		oveq2 6068	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑚 𝑥) = (𝐴 ↑𝑚 𝑦))
8	7	fveq2d 5681	. . . . 5 ⊢ (𝑥 = 𝑦 → (♯‘(𝐴 ↑𝑚 𝑥)) = (♯‘(𝐴 ↑𝑚 𝑦)))
9		fveq2 5677	. . . . . 6 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
10	9	oveq2d 6076	. . . . 5 ⊢ (𝑥 = 𝑦 → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘𝑦)))
11	8, 10	eqeq12d 2249	. . . 4 ⊢ (𝑥 = 𝑦 → ((♯‘(𝐴 ↑𝑚 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴 ↑𝑚 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦))))
12	11	imbi2d 230	. . 3 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)))))
13		oveq2 6068	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ↑𝑚 𝑥) = (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})))
14	13	fveq2d 5681	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘(𝐴 ↑𝑚 𝑥)) = (♯‘(𝐴 ↑𝑚 (𝑦 ∪ {𝑧}))))
15		fveq2 5677	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑧})))
16	15	oveq2d 6076	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))
17	14, 16	eqeq12d 2249	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘(𝐴 ↑𝑚 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴 ↑𝑚 (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧})))))
18	17	imbi2d 230	. . 3 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))))
19		oveq2 6068	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ↑𝑚 𝑥) = (𝐴 ↑𝑚 𝐵))
20	19	fveq2d 5681	. . . . 5 ⊢ (𝑥 = 𝐵 → (♯‘(𝐴 ↑𝑚 𝑥)) = (♯‘(𝐴 ↑𝑚 𝐵)))
21		fveq2 5677	. . . . . 6 ⊢ (𝑥 = 𝐵 → (♯‘𝑥) = (♯‘𝐵))
22	21	oveq2d 6076	. . . . 5 ⊢ (𝑥 = 𝐵 → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘𝐵)))
23	20, 22	eqeq12d 2249	. . . 4 ⊢ (𝑥 = 𝐵 → ((♯‘(𝐴 ↑𝑚 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥)) ↔ (♯‘(𝐴 ↑𝑚 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵))))
24	23	imbi2d 230	. . 3 ⊢ (𝑥 = 𝐵 → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 𝑥)) = ((♯‘𝐴)↑(♯‘𝑥))) ↔ (𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵)))))
25		hashcl 11174	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
26	25	nn0cnd 9577	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℂ)
27	26	exp0d 11059	. . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴)↑0) = 1)
28		hash0 11189	. . . . . 6 ⊢ (♯‘∅) = 0
29	28	oveq2i 6071	. . . . 5 ⊢ ((♯‘𝐴)↑(♯‘∅)) = ((♯‘𝐴)↑0)
30	29	a1i 9	. . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴)↑(♯‘∅)) = ((♯‘𝐴)↑0))
31		mapdm0 6912	. . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ↑𝑚 ∅) = {∅})
32	31	fveq2d 5681	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 ∅)) = (♯‘{∅}))
33		0ex 4243	. . . . . 6 ⊢ ∅ ∈ V
34		hashsng 11191	. . . . . 6 ⊢ (∅ ∈ V → (♯‘{∅}) = 1)
35	33, 34	mp1i 10	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘{∅}) = 1)
36	32, 35	eqtrd 2267	. . . 4 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 ∅)) = 1)
37	27, 30, 36	3eqtr4rd 2278	. . 3 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 ∅)) = ((♯‘𝐴)↑(♯‘∅)))
38		oveq1 6067	. . . . . 6 ⊢ ((♯‘(𝐴 ↑𝑚 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → ((♯‘(𝐴 ↑𝑚 𝑦)) · (♯‘𝐴)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
39		vex 2818	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
40	39	a1i 9	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ V)
41		vsnex 4330	. . . . . . . . . . 11 ⊢ {𝑧} ∈ V
42	41	a1i 9	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → {𝑧} ∈ V)
43		elex 2827	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ V)
44	43	adantr 276	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝐴 ∈ V)
45		simprr 533	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ¬ 𝑧 ∈ 𝑦)
46		disjsn 3757	. . . . . . . . . . 11 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
47	45, 46	sylibr 134	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
48		mapunen 7119	. . . . . . . . . 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 1277	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ≈ ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})))
50		simpl 109	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝐴 ∈ Fin)
51		simprl 531	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ Fin)
52		vex 2818	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
53	52	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑧 ∈ V)
54		unsnfi 7194	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧 ∈ 𝑦) → (𝑦 ∪ {𝑧}) ∈ Fin)
55	51, 53, 45, 54	syl3anc 1274	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑦 ∪ {𝑧}) ∈ Fin)
56		mapfi 7229	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ∈ Fin)
57	50, 55, 56	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ∈ Fin)
58		mapfi 7229	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝐴 ↑𝑚 𝑦) ∈ Fin)
59	58	adantrr 479	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑𝑚 𝑦) ∈ Fin)
60		snfig 7071	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ V → {𝑧} ∈ Fin)
61	60	elv 2819	. . . . . . . . . . . 12 ⊢ {𝑧} ∈ Fin
62		mapfi 7229	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ↑𝑚 {𝑧}) ∈ Fin)
63	50, 61, 62	sylancl 413	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑𝑚 {𝑧}) ∈ Fin)
64		xpfi 7207	. . . . . . . . . . 11 ⊢ (((𝐴 ↑𝑚 𝑦) ∈ Fin ∧ (𝐴 ↑𝑚 {𝑧}) ∈ Fin) → ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})) ∈ Fin)
65	59, 63, 64	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})) ∈ Fin)
66		hashen 11177	. . . . . . . . . 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}))))
67	57, 65, 66	syl2anc 411	. . . . . . . . 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}))))
68	49, 67	mpbird 167	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝐴 ↑𝑚 (𝑦 ∪ {𝑧}))) = (♯‘((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧}))))
69		hashxp 11221	. . . . . . . . 9 ⊢ (((𝐴 ↑𝑚 𝑦) ∈ Fin ∧ (𝐴 ↑𝑚 {𝑧}) ∈ Fin) → (♯‘((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧}))) = ((♯‘(𝐴 ↑𝑚 𝑦)) · (♯‘(𝐴 ↑𝑚 {𝑧}))))
70	59, 63, 69	syl2anc 411	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧}))) = ((♯‘(𝐴 ↑𝑚 𝑦)) · (♯‘(𝐴 ↑𝑚 {𝑧}))))
71	50, 53	mapsnend 7067	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑𝑚 {𝑧}) ≈ 𝐴)
72		hashen 11177	. . . . . . . . . . 11 ⊢ (((𝐴 ↑𝑚 {𝑧}) ∈ Fin ∧ 𝐴 ∈ Fin) → ((♯‘(𝐴 ↑𝑚 {𝑧})) = (♯‘𝐴) ↔ (𝐴 ↑𝑚 {𝑧}) ≈ 𝐴))
73	63, 50, 72	syl2anc 411	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑𝑚 {𝑧})) = (♯‘𝐴) ↔ (𝐴 ↑𝑚 {𝑧}) ≈ 𝐴))
74	71, 73	mpbird 167	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝐴 ↑𝑚 {𝑧})) = (♯‘𝐴))
75	74	oveq2d 6076	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑𝑚 𝑦)) · (♯‘(𝐴 ↑𝑚 {𝑧}))) = ((♯‘(𝐴 ↑𝑚 𝑦)) · (♯‘𝐴)))
76	68, 70, 75	3eqtrd 2271	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝐴 ↑𝑚 (𝑦 ∪ {𝑧}))) = ((♯‘(𝐴 ↑𝑚 𝑦)) · (♯‘𝐴)))
77		hashunsng 11202	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1)))
78	77	elv 2819	. . . . . . . . . 10 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1))
79	78	adantl 277	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1))
80	79	oveq2d 6076	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑((♯‘𝑦) + 1)))
81	26	adantr 276	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘𝐴) ∈ ℂ)
82		hashcl 11174	. . . . . . . . . 10 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ0)
83	82	ad2antrl 490	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘𝑦) ∈ ℕ0)
84	81, 83	expp1d 11066	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘𝐴)↑((♯‘𝑦) + 1)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
85	80, 84	eqtrd 2267	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
86	76, 85	eqeq12d 2249	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑𝑚 (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) ↔ ((♯‘(𝐴 ↑𝑚 𝑦)) · (♯‘𝐴)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴))))
87	38, 86	imbitrrid 156	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑𝑚 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → (♯‘(𝐴 ↑𝑚 (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧})))))
88	87	expcom 116	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝐴 ∈ Fin → ((♯‘(𝐴 ↑𝑚 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → (♯‘(𝐴 ↑𝑚 (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))))
89	88	a2d 26	. . 3 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦))) → (𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 (𝑦 ∪ {𝑧}))) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))))
90	6, 12, 18, 24, 37, 89	findcard2s 7162	. 2 ⊢ (𝐵 ∈ Fin → (𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵))))
91	90	impcom 125	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (♯‘(𝐴 ↑𝑚 𝐵)) = ((♯‘𝐴)↑(♯‘𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  Vcvv 2815   ∪ cun 3212   ∩ cin 3213  ∅c0 3512  {csn 3695   class class class wbr 4115   × cxp 4754  ‘cfv 5359  (class class class)co 6060   ↑_𝑚 cmap 6897   ≈ cen 6988  Fincfn 6990  ℂcc 8143  0cc0 8145  1c1 8146   + caddc 8148   · cmul 8150  ℕ₀cn0 9518  ↑cexp 10929  ♯chash 11168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-coll 4231  ax-sep 4234  ax-nul 4242  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4666  ax-iinf 4717  ax-cnex 8236  ax-resscn 8237  ax-1cn 8238  ax-1re 8239  ax-icn 8240  ax-addcl 8241  ax-addrcl 8242  ax-mulcl 8243  ax-mulrcl 8244  ax-addcom 8245  ax-mulcom 8246  ax-addass 8247  ax-mulass 8248  ax-distr 8249  ax-i2m1 8250  ax-0lt1 8251  ax-1rid 8252  ax-0id 8253  ax-rnegex 8254  ax-precex 8255  ax-cnre 8256  ax-pre-ltirr 8257  ax-pre-ltwlin 8258  ax-pre-lttrn 8259  ax-pre-apti 8260  ax-pre-ltadd 8261  ax-pre-mulgt0 8262  ax-pre-mulext 8263

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3626  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-iun 3999  df-br 4116  df-opab 4178  df-mpt 4179  df-tr 4215  df-id 4420  df-po 4423  df-iso 4424  df-iord 4493  df-on 4495  df-ilim 4496  df-suc 4498  df-iom 4720  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-ima 4769  df-iota 5319  df-fun 5361  df-fn 5362  df-f 5363  df-f1 5364  df-fo 5365  df-f1o 5366  df-fv 5367  df-riota 6013  df-ov 6063  df-oprab 6064  df-mpo 6065  df-1st 6349  df-2nd 6350  df-recs 6551  df-irdg 6616  df-frec 6637  df-1o 6662  df-oadd 6666  df-er 6782  df-map 6899  df-en 6991  df-dom 6992  df-fin 6993  df-pnf 8328  df-mnf 8329  df-xr 8330  df-ltxr 8331  df-le 8332  df-sub 8465  df-neg 8466  df-reap 8869  df-ap 8876  df-div 8969  df-inn 9260  df-n0 9519  df-z 9600  df-uz 9877  df-fz 10367  df-seqfrec 10839  df-exp 10930  df-ihash 11169

This theorem is referenced by:  hashpwfi  11223