Theorem hashmap 11246

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 6083	. . . . . 6 ⊢ (𝑥 = ∅ → (𝐴 ↑𝑚 𝑥) = (𝐴 ↑𝑚 ∅))
2	1	fveq2d 5694	. . . . 5 ⊢ (𝑥 = ∅ → (♯‘(𝐴 ↑𝑚 𝑥)) = (♯‘(𝐴 ↑𝑚 ∅)))
3		fveq2 5690	. . . . . 6 ⊢ (𝑥 = ∅ → (♯‘𝑥) = (♯‘∅))
4	3	oveq2d 6091	. . . . 5 ⊢ (𝑥 = ∅ → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘∅)))
5	2, 4	eqeq12d 2253	. . . 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 6083	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴 ↑𝑚 𝑥) = (𝐴 ↑𝑚 𝑦))
8	7	fveq2d 5694	. . . . 5 ⊢ (𝑥 = 𝑦 → (♯‘(𝐴 ↑𝑚 𝑥)) = (♯‘(𝐴 ↑𝑚 𝑦)))
9		fveq2 5690	. . . . . 6 ⊢ (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
10	9	oveq2d 6091	. . . . 5 ⊢ (𝑥 = 𝑦 → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘𝑦)))
11	8, 10	eqeq12d 2253	. . . 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 6083	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ↑𝑚 𝑥) = (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})))
14	13	fveq2d 5694	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘(𝐴 ↑𝑚 𝑥)) = (♯‘(𝐴 ↑𝑚 (𝑦 ∪ {𝑧}))))
15		fveq2 5690	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (♯‘𝑥) = (♯‘(𝑦 ∪ {𝑧})))
16	15	oveq2d 6091	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))))
17	14, 16	eqeq12d 2253	. . . 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 6083	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝐴 ↑𝑚 𝑥) = (𝐴 ↑𝑚 𝐵))
20	19	fveq2d 5694	. . . . 5 ⊢ (𝑥 = 𝐵 → (♯‘(𝐴 ↑𝑚 𝑥)) = (♯‘(𝐴 ↑𝑚 𝐵)))
21		fveq2 5690	. . . . . 6 ⊢ (𝑥 = 𝐵 → (♯‘𝑥) = (♯‘𝐵))
22	21	oveq2d 6091	. . . . 5 ⊢ (𝑥 = 𝐵 → ((♯‘𝐴)↑(♯‘𝑥)) = ((♯‘𝐴)↑(♯‘𝐵)))
23	20, 22	eqeq12d 2253	. . . 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 11198	. . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0)
26	25	nn0cnd 9601	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℂ)
27	26	exp0d 11083	. . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴)↑0) = 1)
28		hash0 11213	. . . . . 6 ⊢ (♯‘∅) = 0
29	28	oveq2i 6086	. . . . 5 ⊢ ((♯‘𝐴)↑(♯‘∅)) = ((♯‘𝐴)↑0)
30	29	a1i 9	. . . 4 ⊢ (𝐴 ∈ Fin → ((♯‘𝐴)↑(♯‘∅)) = ((♯‘𝐴)↑0))
31		mapdm0 6927	. . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ↑𝑚 ∅) = {∅})
32	31	fveq2d 5694	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 ∅)) = (♯‘{∅}))
33		0ex 4255	. . . . . 6 ⊢ ∅ ∈ V
34		hashsng 11215	. . . . . 6 ⊢ (∅ ∈ V → (♯‘{∅}) = 1)
35	33, 34	mp1i 10	. . . . 5 ⊢ (𝐴 ∈ Fin → (♯‘{∅}) = 1)
36	32, 35	eqtrd 2271	. . . 4 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 ∅)) = 1)
37	27, 30, 36	3eqtr4rd 2282	. . 3 ⊢ (𝐴 ∈ Fin → (♯‘(𝐴 ↑𝑚 ∅)) = ((♯‘𝐴)↑(♯‘∅)))
38		oveq1 6082	. . . . . 6 ⊢ ((♯‘(𝐴 ↑𝑚 𝑦)) = ((♯‘𝐴)↑(♯‘𝑦)) → ((♯‘(𝐴 ↑𝑚 𝑦)) · (♯‘𝐴)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
39		vex 2824	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
40	39	a1i 9	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ V)
41		vsnex 4343	. . . . . . . . . . 11 ⊢ {𝑧} ∈ V
42	41	a1i 9	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → {𝑧} ∈ V)
43		elex 2833	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ V)
44	43	adantr 276	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝐴 ∈ V)
45		simprr 537	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ¬ 𝑧 ∈ 𝑦)
46		disjsn 3767	. . . . . . . . . . 11 ⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦)
47	45, 46	sylibr 134	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅)
48		mapunen 7141	. . . . . . . . . 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 1281	. . . . . . . . 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 535	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ Fin)
52		vex 2824	. . . . . . . . . . . . 13 ⊢ 𝑧 ∈ V
53	52	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑧 ∈ V)
54		unsnfi 7216	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧 ∈ 𝑦) → (𝑦 ∪ {𝑧}) ∈ Fin)
55	51, 53, 45, 54	syl3anc 1278	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑦 ∪ {𝑧}) ∈ Fin)
56		mapfi 7251	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ∈ Fin)
57	50, 55, 56	syl2anc 415	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑𝑚 (𝑦 ∪ {𝑧})) ∈ Fin)
58		mapfi 7251	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑦 ∈ Fin) → (𝐴 ↑𝑚 𝑦) ∈ Fin)
59	58	adantrr 483	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑𝑚 𝑦) ∈ Fin)
60		snfig 7093	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ V → {𝑧} ∈ Fin)
61	60	elv 2825	. . . . . . . . . . . 12 ⊢ {𝑧} ∈ Fin
62		mapfi 7251	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝐴 ↑𝑚 {𝑧}) ∈ Fin)
63	50, 61, 62	sylancl 417	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑𝑚 {𝑧}) ∈ Fin)
64		xpfi 7229	. . . . . . . . . . 11 ⊢ (((𝐴 ↑𝑚 𝑦) ∈ Fin ∧ (𝐴 ↑𝑚 {𝑧}) ∈ Fin) → ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})) ∈ Fin)
65	59, 63, 64	syl2anc 415	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧})) ∈ Fin)
66		hashen 11201	. . . . . . . . . 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 415	. . . . . . . . 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 11245	. . . . . . . . 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 415	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘((𝐴 ↑𝑚 𝑦) × (𝐴 ↑𝑚 {𝑧}))) = ((♯‘(𝐴 ↑𝑚 𝑦)) · (♯‘(𝐴 ↑𝑚 {𝑧}))))
71	50, 53	mapsnend 7089	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↑𝑚 {𝑧}) ≈ 𝐴)
72		hashen 11201	. . . . . . . . . . 11 ⊢ (((𝐴 ↑𝑚 {𝑧}) ∈ Fin ∧ 𝐴 ∈ Fin) → ((♯‘(𝐴 ↑𝑚 {𝑧})) = (♯‘𝐴) ↔ (𝐴 ↑𝑚 {𝑧}) ≈ 𝐴))
73	63, 50, 72	syl2anc 415	. . . . . . . . . 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 6091	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘(𝐴 ↑𝑚 𝑦)) · (♯‘(𝐴 ↑𝑚 {𝑧}))) = ((♯‘(𝐴 ↑𝑚 𝑦)) · (♯‘𝐴)))
76	68, 70, 75	3eqtrd 2275	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘(𝐴 ↑𝑚 (𝑦 ∪ {𝑧}))) = ((♯‘(𝐴 ↑𝑚 𝑦)) · (♯‘𝐴)))
77		hashunsng 11226	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (♯‘(𝑦 ∪ {𝑧})) = ((♯‘𝑦) + 1)))
78	77	elv 2825	. . . . . . . . . 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 6091	. . . . . . . 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 11198	. . . . . . . . . 10 ⊢ (𝑦 ∈ Fin → (♯‘𝑦) ∈ ℕ0)
83	82	ad2antrl 494	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (♯‘𝑦) ∈ ℕ0)
84	81, 83	expp1d 11090	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘𝐴)↑((♯‘𝑦) + 1)) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
85	80, 84	eqtrd 2271	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((♯‘𝐴)↑(♯‘(𝑦 ∪ {𝑧}))) = (((♯‘𝐴)↑(♯‘𝑦)) · (♯‘𝐴)))
86	76, 85	eqeq12d 2253	. . . . . 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 7184	. 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 1402   ∈ wcel 2209  Vcvv 2821   ∪ cun 3218   ∩ cin 3219  ∅c0 3520  {csn 3705   class class class wbr 4125   × cxp 4767  ‘cfv 5372  (class class class)co 6075   ↑_𝑚 cmap 6912   ≈ cen 7010  Fincfn 7012  ℂcc 8167  0cc0 8169  1c1 8170   + caddc 8172   · cmul 8174  ℕ₀cn0 9542  ↑cexp 10953  ♯chash 11192

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-map 6914  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-seqfrec 10863  df-exp 10954  df-ihash 11193

This theorem is referenced by:  hashpwfi  11247