Theorem mapunen 7141

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 6919	. . 3 ⊢ ↑𝑚 Fn (V × V)
2		simpl3 1033	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐶 ∈ 𝑋)
3	2	elexd 2835	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐶 ∈ V)
4		simpl1 1031	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ 𝑉)
5		simpl2 1032	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ 𝑊)
6	4, 5	unexd 4887	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ V)
7		fnovex 6108	. . 3 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐶 ∈ V ∧ (𝐴 ∪ 𝐵) ∈ V) → (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∈ V)
8	1, 3, 6, 7	mp3an2i 1383	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∈ V)
9	4	elexd 2835	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ V)
10		fnovex 6108	. . . 4 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐶 ∈ V ∧ 𝐴 ∈ V) → (𝐶 ↑𝑚 𝐴) ∈ V)
11	1, 3, 9, 10	mp3an2i 1383	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑𝑚 𝐴) ∈ V)
12	5	elexd 2835	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ V)
13		fnovex 6108	. . . 4 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐶 ∈ V ∧ 𝐵 ∈ V) → (𝐶 ↑𝑚 𝐵) ∈ V)
14	1, 3, 12, 13	mp3an2i 1383	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑𝑚 𝐵) ∈ V)
15	11, 14	xpexd 4885	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) ∈ V)
16		elmapi 6934	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → 𝑥:(𝐴 ∪ 𝐵)⟶𝐶)
17		ssun1 3392	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
18		fssres 5560	. . . . 5 ⊢ ((𝑥:(𝐴 ∪ 𝐵)⟶𝐶 ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐴):𝐴⟶𝐶)
19	16, 17, 18	sylancl 417	. . . 4 ⊢ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐴):𝐴⟶𝐶)
20		ssun2 3393	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
21		fssres 5560	. . . . 5 ⊢ ((𝑥:(𝐴 ∪ 𝐵)⟶𝐶 ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐵):𝐵⟶𝐶)
22	16, 20, 21	sylancl 417	. . . 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 4799	. . . 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 6926	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ↾ 𝐴) ∈ (𝐶 ↑𝑚 𝐴) ↔ (𝑥 ↾ 𝐴):𝐴⟶𝐶))
26	2, 5	elmapd 6926	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ↾ 𝐵) ∈ (𝐶 ↑𝑚 𝐵) ↔ (𝑥 ↾ 𝐵):𝐵⟶𝐶))
27	25, 26	anbi12d 477	. . . 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 6389	. . . . . . 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 6934	. . . . . 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 6390	. . . . . . 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 6934	. . . . . 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 533	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → (𝐴 ∩ 𝐵) = ∅)
39	33, 37, 38	fun2d 5558	. . . 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 6926	. . 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 6399	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
44	43	ad2antll 495	. . . . . 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 482	. . . . . . . 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 482	. . . . . . . 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 533	. . . . . . . 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 5566	. . . . . . . 8 ⊢ (((1st ‘𝑦):𝐴⟶𝐶 ∧ (2nd ‘𝑦):𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴) = (1st ‘𝑦))
49	45, 46, 47, 48	syl3anc 1278	. . . . . . 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 5565	. . . . . . . 8 ⊢ (((1st ‘𝑦):𝐴⟶𝐶 ∧ (2nd ‘𝑦):𝐵⟶𝐶 ∧ (𝐴 ∩ 𝐵) = ∅) → (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵) = (2nd ‘𝑦))
51	45, 46, 47, 50	syl3anc 1278	. . . . . . 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 3907	. . . . . 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 2274	. . . . 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 5052	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → (𝑥 ↾ 𝐴) = (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴))
55		reseq1 5052	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → (𝑥 ↾ 𝐵) = (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵))
56	54, 55	opeq12d 3907	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ = ⟨(((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴), (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵)⟩)
57	56	eqeq2d 2250	. . . . 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 5528	. . . . . . . 8 ⊢ (𝑥:(𝐴 ∪ 𝐵)⟶𝐶 → 𝑥 Fn (𝐴 ∪ 𝐵))
60		fnresdm 5487	. . . . . . . 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 494	. . . . . 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 2244	. . . . 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 2824	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
65	64	resex 5099	. . . . . . . . 9 ⊢ (𝑥 ↾ 𝐴) ∈ V
66	64	resex 5099	. . . . . . . . 9 ⊢ (𝑥 ↾ 𝐵) ∈ V
67	65, 66	op1std 6372	. . . . . . . 8 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (1st ‘𝑦) = (𝑥 ↾ 𝐴))
68	65, 66	op2ndd 6373	. . . . . . . 8 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (2nd ‘𝑦) = (𝑥 ↾ 𝐵))
69	67, 68	uneq12d 3384	. . . . . . 7 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) = ((𝑥 ↾ 𝐴) ∪ (𝑥 ↾ 𝐵)))
70		resundi 5071	. . . . . . 7 ⊢ (𝑥 ↾ (𝐴 ∪ 𝐵)) = ((𝑥 ↾ 𝐴) ∪ (𝑥 ↾ 𝐵))
71	69, 70	eqtr4di 2289	. . . . . 6 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) = (𝑥 ↾ (𝐴 ∪ 𝐵)))
72	71	eqeq2d 2250	. . . . 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 7045	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 1009   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∪ cun 3218   ∩ cin 3219   ⊆ wss 3220  ∅c0 3520  ⟨cop 3708   class class class wbr 4125   × cxp 4767   ↾ cres 4771   Fn wfn 5367  ⟶wf 5368  ‘cfv 5372  (class class class)co 6075  1^st c1st 6362  2^nd c2nd 6363   ↑_𝑚 cmap 6912   ≈ cen 7010

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-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  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-ral 2533  df-rex 2534  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-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  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-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-en 7013

This theorem is referenced by:  mapfi  7251  hashmap  11246