Theorem mapunen 8129

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		ovex 6678	. . 3 ⊢ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∈ V
2	1	a1i 11	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∈ V)
3		ovex 6678	. . . 4 ⊢ (𝐶 ↑𝑚 𝐴) ∈ V
4		ovex 6678	. . . 4 ⊢ (𝐶 ↑𝑚 𝐵) ∈ V
5	3, 4	xpex 6962	. . 3 ⊢ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) ∈ V
6	5	a1i 11	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) ∈ V)
7		elmapi 7879	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → 𝑥:(𝐴 ∪ 𝐵)⟶𝐶)
8		ssun1 3776	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
9		fssres 6070	. . . . 5 ⊢ ((𝑥:(𝐴 ∪ 𝐵)⟶𝐶 ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐴):𝐴⟶𝐶)
10	7, 8, 9	sylancl 694	. . . 4 ⊢ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐴):𝐴⟶𝐶)
11		ssun2 3777	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
12		fssres 6070	. . . . 5 ⊢ ((𝑥:(𝐴 ∪ 𝐵)⟶𝐶 ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐵):𝐵⟶𝐶)
13	7, 11, 12	sylancl 694	. . . 4 ⊢ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐵):𝐵⟶𝐶)
14	10, 13	jca 554	. . 3 ⊢ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → ((𝑥 ↾ 𝐴):𝐴⟶𝐶 ∧ (𝑥 ↾ 𝐵):𝐵⟶𝐶))
15		opelxp 5146	. . . 4 ⊢ (⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) ↔ ((𝑥 ↾ 𝐴) ∈ (𝐶 ↑𝑚 𝐴) ∧ (𝑥 ↾ 𝐵) ∈ (𝐶 ↑𝑚 𝐵)))
16		simpl3 1066	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐶 ∈ 𝑋)
17		simpl1 1064	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ 𝑉)
18	16, 17	elmapd 7871	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ↾ 𝐴) ∈ (𝐶 ↑𝑚 𝐴) ↔ (𝑥 ↾ 𝐴):𝐴⟶𝐶))
19		simpl2 1065	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ 𝑊)
20	16, 19	elmapd 7871	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ↾ 𝐵) ∈ (𝐶 ↑𝑚 𝐵) ↔ (𝑥 ↾ 𝐵):𝐵⟶𝐶))
21	18, 20	anbi12d 747	. . . 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)))
22	15, 21	syl5bb 272	. . 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)))
23	14, 22	syl5ibr 236	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))))
24		xp1st 7198	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) → (1st ‘𝑦) ∈ (𝐶 ↑𝑚 𝐴))
25	24	adantl 482	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → (1st ‘𝑦) ∈ (𝐶 ↑𝑚 𝐴))
26		elmapi 7879	. . . . . 6 ⊢ ((1st ‘𝑦) ∈ (𝐶 ↑𝑚 𝐴) → (1st ‘𝑦):𝐴⟶𝐶)
27	25, 26	syl 17	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → (1st ‘𝑦):𝐴⟶𝐶)
28		xp2nd 7199	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) → (2nd ‘𝑦) ∈ (𝐶 ↑𝑚 𝐵))
29	28	adantl 482	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → (2nd ‘𝑦) ∈ (𝐶 ↑𝑚 𝐵))
30		elmapi 7879	. . . . . 6 ⊢ ((2nd ‘𝑦) ∈ (𝐶 ↑𝑚 𝐵) → (2nd ‘𝑦):𝐵⟶𝐶)
31	29, 30	syl 17	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → (2nd ‘𝑦):𝐵⟶𝐶)
32		simplr 792	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → (𝐴 ∩ 𝐵) = ∅)
33		fun2 6067	. . . . 5 ⊢ ((((1st ‘𝑦):𝐴⟶𝐶 ∧ (2nd ‘𝑦):𝐵⟶𝐶) ∧ (𝐴 ∩ 𝐵) = ∅) → ((1st ‘𝑦) ∪ (2nd ‘𝑦)):(𝐴 ∪ 𝐵)⟶𝐶)
34	27, 31, 32, 33	syl21anc 1325	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → ((1st ‘𝑦) ∪ (2nd ‘𝑦)):(𝐴 ∪ 𝐵)⟶𝐶)
35	34	ex 450	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) → ((1st ‘𝑦) ∪ (2nd ‘𝑦)):(𝐴 ∪ 𝐵)⟶𝐶))
36		unexg 6959	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
37	17, 19, 36	syl2anc 693	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ V)
38	16, 37	elmapd 7871	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ↔ ((1st ‘𝑦) ∪ (2nd ‘𝑦)):(𝐴 ∪ 𝐵)⟶𝐶))
39	35, 38	sylibrd 249	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵))))
40		1st2nd2 7205	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
41	40	ad2antll 765	. . . . . 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)⟩)
42	27	adantrl 752	. . . . . . . 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)
43	31	adantrl 752	. . . . . . . 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)
44		res0 5400	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ↾ ∅) = ∅
45		res0 5400	. . . . . . . . . 10 ⊢ ((2nd ‘𝑦) ↾ ∅) = ∅
46	44, 45	eqtr4i 2647	. . . . . . . . 9 ⊢ ((1st ‘𝑦) ↾ ∅) = ((2nd ‘𝑦) ↾ ∅)
47		simplr 792	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → (𝐴 ∩ 𝐵) = ∅)
48	47	reseq2d 5396	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → ((1st ‘𝑦) ↾ (𝐴 ∩ 𝐵)) = ((1st ‘𝑦) ↾ ∅))
49	47	reseq2d 5396	. . . . . . . . 9 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → ((2nd ‘𝑦) ↾ (𝐴 ∩ 𝐵)) = ((2nd ‘𝑦) ↾ ∅))
50	46, 48, 49	3eqtr4a 2682	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))) → ((1st ‘𝑦) ↾ (𝐴 ∩ 𝐵)) = ((2nd ‘𝑦) ↾ (𝐴 ∩ 𝐵)))
51		fresaunres1 6077	. . . . . . . 8 ⊢ (((1st ‘𝑦):𝐴⟶𝐶 ∧ (2nd ‘𝑦):𝐵⟶𝐶 ∧ ((1st ‘𝑦) ↾ (𝐴 ∩ 𝐵)) = ((2nd ‘𝑦) ↾ (𝐴 ∩ 𝐵))) → (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴) = (1st ‘𝑦))
52	42, 43, 50, 51	syl3anc 1326	. . . . . . 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))
53		fresaunres2 6076	. . . . . . . 8 ⊢ (((1st ‘𝑦):𝐴⟶𝐶 ∧ (2nd ‘𝑦):𝐵⟶𝐶 ∧ ((1st ‘𝑦) ↾ (𝐴 ∩ 𝐵)) = ((2nd ‘𝑦) ↾ (𝐴 ∩ 𝐵))) → (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵) = (2nd ‘𝑦))
54	42, 43, 50, 53	syl3anc 1326	. . . . . . 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))
55	52, 54	opeq12d 4410	. . . . . 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)⟩)
56	41, 55	eqtr4d 2659	. . . . 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)⟩)
57		reseq1 5390	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → (𝑥 ↾ 𝐴) = (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴))
58		reseq1 5390	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → (𝑥 ↾ 𝐵) = (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵))
59	57, 58	opeq12d 4410	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ = ⟨(((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴), (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵)⟩)
60	59	eqeq2d 2632	. . . . 5 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ ↔ 𝑦 = ⟨(((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴), (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵)⟩))
61	56, 60	syl5ibrcom 237	. . . 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)⟩))
62		ffn 6045	. . . . . . . 8 ⊢ (𝑥:(𝐴 ∪ 𝐵)⟶𝐶 → 𝑥 Fn (𝐴 ∪ 𝐵))
63		fnresdm 6000	. . . . . . . 8 ⊢ (𝑥 Fn (𝐴 ∪ 𝐵) → (𝑥 ↾ (𝐴 ∪ 𝐵)) = 𝑥)
64	7, 62, 63	3syl 18	. . . . . . 7 ⊢ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → (𝑥 ↾ (𝐴 ∪ 𝐵)) = 𝑥)
65	64	ad2antrl 764	. . . . . 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)
66	65	eqcomd 2628	. . . . 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)))
67		vex 3203	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
68	67	resex 5443	. . . . . . . . 9 ⊢ (𝑥 ↾ 𝐴) ∈ V
69	67	resex 5443	. . . . . . . . 9 ⊢ (𝑥 ↾ 𝐵) ∈ V
70	68, 69	op1std 7178	. . . . . . . 8 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (1st ‘𝑦) = (𝑥 ↾ 𝐴))
71	68, 69	op2ndd 7179	. . . . . . . 8 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (2nd ‘𝑦) = (𝑥 ↾ 𝐵))
72	70, 71	uneq12d 3768	. . . . . . 7 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) = ((𝑥 ↾ 𝐴) ∪ (𝑥 ↾ 𝐵)))
73		resundi 5410	. . . . . . 7 ⊢ (𝑥 ↾ (𝐴 ∪ 𝐵)) = ((𝑥 ↾ 𝐴) ∪ (𝑥 ↾ 𝐵))
74	72, 73	syl6eqr 2674	. . . . . 6 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) = (𝑥 ↾ (𝐴 ∪ 𝐵)))
75	74	eqeq2d 2632	. . . . 5 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↔ 𝑥 = (𝑥 ↾ (𝐴 ∪ 𝐵))))
76	66, 75	syl5ibrcom 237	. . . 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))))
77	61, 76	impbid 202	. . 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)⟩))
78	77	ex 450	. 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)⟩)))
79	2, 6, 23, 39, 78	en3d 7992	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ≈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ⟨cop 4183   class class class wbr 4653   × cxp 5112   ↾ cres 5116   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167   ↑_𝑚 cmap 7857   ≈ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-en 7956

This theorem is referenced by:  map2xp  8130  mapdom2  8131  mapcdaen  9006  ackbij1lem5  9046  hashmap  13222  mpct  39393