Theorem mapunen 7095

Description: Equinumerosity law for set exponentiation of a disjoint union. Exercise 4.45 of [Mendelson] p. 255. (Contributed by NM, 23-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
mapunen	⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ≈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))

Proof of Theorem mapunen

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnmap 6880	. . 3 ⊢ ↑𝑚 Fn (V × V)
2		simpl3 1029	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐶 ∈ 𝑋)
3	2	elexd 2826	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐶 ∈ V)
4		simpl1 1027	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ 𝑉)
5		simpl2 1028	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ 𝑊)
6	4, 5	unexd 4858	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ V)
7		fnovex 6074	. . 3 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐶 ∈ V ∧ (𝐴 ∪ 𝐵) ∈ V) → (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∈ V)
8	1, 3, 6, 7	mp3an2i 1379	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∈ V)
9	4	elexd 2826	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ V)
10		fnovex 6074	. . . 4 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐶 ∈ V ∧ 𝐴 ∈ V) → (𝐶 ↑𝑚 𝐴) ∈ V)
11	1, 3, 9, 10	mp3an2i 1379	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑𝑚 𝐴) ∈ V)
12	5	elexd 2826	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ V)
13		fnovex 6074	. . . 4 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐶 ∈ V ∧ 𝐵 ∈ V) → (𝐶 ↑𝑚 𝐵) ∈ V)
14	1, 3, 12, 13	mp3an2i 1379	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑𝑚 𝐵) ∈ V)
15	11, 14	xpexd 4856	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) ∈ V)
16		elmapi 6895	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → 𝑥:(𝐴 ∪ 𝐵)⟶𝐶)
17		ssun1 3381	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
18		fssres 5531	. . . . 5 ⊢ ((𝑥:(𝐴 ∪ 𝐵)⟶𝐶 ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐴):𝐴⟶𝐶)
19	16, 17, 18	sylancl 413	. . . 4 ⊢ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐴):𝐴⟶𝐶)
20		ssun2 3382	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
21		fssres 5531	. . . . 5 ⊢ ((𝑥:(𝐴 ∪ 𝐵)⟶𝐶 ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐵):𝐵⟶𝐶)
22	16, 20, 21	sylancl 413	. . . 4 ⊢ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐵):𝐵⟶𝐶)
23	19, 22	jca 306	. . 3 ⊢ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → ((𝑥 ↾ 𝐴):𝐴⟶𝐶 ∧ (𝑥 ↾ 𝐵):𝐵⟶𝐶))
24		opelxp 4770	. . . 4 ⊢ (⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) ↔ ((𝑥 ↾ 𝐴) ∈ (𝐶 ↑𝑚 𝐴) ∧ (𝑥 ↾ 𝐵) ∈ (𝐶 ↑𝑚 𝐵)))
25	2, 4	elmapd 6887	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ↾ 𝐴) ∈ (𝐶 ↑𝑚 𝐴) ↔ (𝑥 ↾ 𝐴):𝐴⟶𝐶))
26	2, 5	elmapd 6887	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ↾ 𝐵) ∈ (𝐶 ↑𝑚 𝐵) ↔ (𝑥 ↾ 𝐵):𝐵⟶𝐶))
27	25, 26	anbi12d 473	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (((𝑥 ↾ 𝐴) ∈ (𝐶 ↑𝑚 𝐴) ∧ (𝑥 ↾ 𝐵) ∈ (𝐶 ↑𝑚 𝐵)) ↔ ((𝑥 ↾ 𝐴):𝐴⟶𝐶 ∧ (𝑥 ↾ 𝐵):𝐵⟶𝐶)))
28	24, 27	bitrid 192	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) ↔ ((𝑥 ↾ 𝐴):𝐴⟶𝐶 ∧ (𝑥 ↾ 𝐵):𝐵⟶𝐶)))
29	23, 28	imbitrrid 156	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))))
30		xp1st 6350	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) → (1st ‘𝑦) ∈ (𝐶 ↑𝑚 𝐴))
31	30	adantl 277	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → (1st ‘𝑦) ∈ (𝐶 ↑𝑚 𝐴))
32		elmapi 6895	. . . . . 6 ⊢ ((1st ‘𝑦) ∈ (𝐶 ↑𝑚 𝐴) → (1st ‘𝑦):𝐴⟶𝐶)
33	31, 32	syl 14	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → (1st ‘𝑦):𝐴⟶𝐶)
34		xp2nd 6351	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) → (2nd ‘𝑦) ∈ (𝐶 ↑𝑚 𝐵))
35	34	adantl 277	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → (2nd ‘𝑦) ∈ (𝐶 ↑𝑚 𝐵))
36		elmapi 6895	. . . . . 6 ⊢ ((2nd ‘𝑦) ∈ (𝐶 ↑𝑚 𝐵) → (2nd ‘𝑦):𝐵⟶𝐶)
37	35, 36	syl 14	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → (2nd ‘𝑦):𝐵⟶𝐶)
38		simplr 529	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → (𝐴 ∩ 𝐵) = ∅)
39	33, 37, 38	fun2d 5529	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → ((1st ‘𝑦) ∪ (2nd ‘𝑦)):(𝐴 ∪ 𝐵)⟶𝐶)
40	39	ex 115	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) → ((1st ‘𝑦) ∪ (2nd ‘𝑦)):(𝐴 ∪ 𝐵)⟶𝐶))
41	2, 6	elmapd 6887	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ↔ ((1st ‘𝑦) ∪ (2nd ‘𝑦)):(𝐴 ∪ 𝐵)⟶𝐶))
42	40, 41	sylibrd 169	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵))))
43		1st2nd2 6360	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
44	43	ad2antll 491	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
45	33	adantrl 478	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → (1st ‘𝑦):𝐴⟶𝐶)
46	37	adantrl 478	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → (2nd ‘𝑦):𝐵⟶𝐶)
47		simplr 529	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → (𝐴 ∩ 𝐵) = ∅)
48		fresaunres1disj 5537	. . . . . . . 8 ⊢ (((1st ‘𝑦):𝐴⟶𝐶 ∧ (2nd ‘𝑦):𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴) = (1st ‘𝑦))
49	45, 46, 47, 48	syl3anc 1274	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴) = (1st ‘𝑦))
50		fresaunres2disj 5536	. . . . . . . 8 ⊢ (((1st ‘𝑦):𝐴⟶𝐶 ∧ (2nd ‘𝑦):𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵) = (2nd ‘𝑦))
51	45, 46, 47, 50	syl3anc 1274	. . . . . . 7 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵) = (2nd ‘𝑦))
52	49, 51	opeq12d 3884	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → ⟨(((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴), (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵)⟩ = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
53	44, 52	eqtr4d 2268	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → 𝑦 = ⟨(((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴), (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵)⟩)
54		reseq1 5023	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → (𝑥 ↾ 𝐴) = (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴))
55		reseq1 5023	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → (𝑥 ↾ 𝐵) = (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵))
56	54, 55	opeq12d 3884	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ = ⟨(((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴), (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵)⟩)
57	56	eqeq2d 2244	. . . . 5 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ ↔ 𝑦 = ⟨(((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴), (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵)⟩))
58	53, 57	syl5ibrcom 157	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → 𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩))
59		ffn 5499	. . . . . . . 8 ⊢ (𝑥:(𝐴 ∪ 𝐵)⟶𝐶 → 𝑥 Fn (𝐴 ∪ 𝐵))
60		fnresdm 5458	. . . . . . . 8 ⊢ (𝑥 Fn (𝐴 ∪ 𝐵) → (𝑥 ↾ (𝐴 ∪ 𝐵)) = 𝑥)
61	16, 59, 60	3syl 17	. . . . . . 7 ⊢ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → (𝑥 ↾ (𝐴 ∪ 𝐵)) = 𝑥)
62	61	ad2antrl 490	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → (𝑥 ↾ (𝐴 ∪ 𝐵)) = 𝑥)
63	62	eqcomd 2238	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → 𝑥 = (𝑥 ↾ (𝐴 ∪ 𝐵)))
64		vex 2815	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
65	64	resex 5070	. . . . . . . . 9 ⊢ (𝑥 ↾ 𝐴) ∈ V
66	64	resex 5070	. . . . . . . . 9 ⊢ (𝑥 ↾ 𝐵) ∈ V
67	65, 66	op1std 6333	. . . . . . . 8 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (1st ‘𝑦) = (𝑥 ↾ 𝐴))
68	65, 66	op2ndd 6334	. . . . . . . 8 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (2nd ‘𝑦) = (𝑥 ↾ 𝐵))
69	67, 68	uneq12d 3373	. . . . . . 7 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) = ((𝑥 ↾ 𝐴) ∪ (𝑥 ↾ 𝐵)))
70		resundi 5042	. . . . . . 7 ⊢ (𝑥 ↾ (𝐴 ∪ 𝐵)) = ((𝑥 ↾ 𝐴) ∪ (𝑥 ↾ 𝐵))
71	69, 70	eqtr4di 2283	. . . . . 6 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) = (𝑥 ↾ (𝐴 ∪ 𝐵)))
72	71	eqeq2d 2244	. . . . 5 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↔ 𝑥 = (𝑥 ↾ (𝐴 ∪ 𝐵))))
73	63, 72	syl5ibrcom 157	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → 𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦))))
74	58, 73	impbid 129	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↔ 𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩))
75	74	ex 115	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↔ 𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩)))
76	8, 15, 29, 42, 75	en3d 6999	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ≈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  Vcvv 2812   ∪ cun 3208   ∩ cin 3209   ⊆ wss 3210  ∅c0 3505  ⟨cop 3685   class class class wbr 4102   × cxp 4738   ↾ cres 4742   Fn wfn 5338  ⟶wf 5339  ‘cfv 5343  (class class class)co 6041  1^st c1st 6323  2^nd c2nd 6324   ↑_𝑚 cmap 6873   ≈ cen 6964

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-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-pow 4279  ax-pr 4314  ax-un 4545  ax-setind 4650

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3506  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-iun 3986  df-br 4103  df-opab 4165  df-mpt 4166  df-id 4405  df-xp 4746  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-rn 4751  df-res 4752  df-ima 4753  df-iota 5303  df-fun 5345  df-fn 5346  df-f 5347  df-f1 5348  df-fo 5349  df-f1o 5350  df-fv 5351  df-ov 6044  df-oprab 6045  df-mpo 6046  df-1st 6325  df-2nd 6326  df-map 6875  df-en 6967

This theorem is referenced by:  mapfi  7205