Theorem mapunen 9172

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		ovexd 7459	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑m (𝐴 ∪ 𝐵)) ∈ V)
2		ovexd 7459	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑m 𝐴) ∈ V)
3		ovexd 7459	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑m 𝐵) ∈ V)
4	2, 3	xpexd 7769	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) ∈ V)
5		elmapi 8876	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → 𝑥:(𝐴 ∪ 𝐵)⟶𝐶)
6		ssun1 4175	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
7		fssres 6753	. . . . 5 ⊢ ((𝑥:(𝐴 ∪ 𝐵)⟶𝐶 ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐴):𝐴⟶𝐶)
8	5, 6, 7	sylancl 598	. . . 4 ⊢ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐴):𝐴⟶𝐶)
9		ssun2 4176	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
10		fssres 6753	. . . . 5 ⊢ ((𝑥:(𝐴 ∪ 𝐵)⟶𝐶 ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐵):𝐵⟶𝐶)
11	5, 9, 10	sylancl 598	. . . 4 ⊢ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐵):𝐵⟶𝐶)
12	8, 11	jca 521	. . 3 ⊢ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → ((𝑥 ↾ 𝐴):𝐴⟶𝐶 ∧ (𝑥 ↾ 𝐵):𝐵⟶𝐶))
13		opelxp 5687	. . . 4 ⊢ (⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) ↔ ((𝑥 ↾ 𝐴) ∈ (𝐶 ↑m 𝐴) ∧ (𝑥 ↾ 𝐵) ∈ (𝐶 ↑m 𝐵)))
14		simpl3 1212	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐶 ∈ 𝑋)
15		simpl1 1210	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ 𝑉)
16	14, 15	elmapd 8867	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ↾ 𝐴) ∈ (𝐶 ↑m 𝐴) ↔ (𝑥 ↾ 𝐴):𝐴⟶𝐶))
17		simpl2 1211	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ 𝑊)
18	14, 17	elmapd 8867	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ↾ 𝐵) ∈ (𝐶 ↑m 𝐵) ↔ (𝑥 ↾ 𝐵):𝐵⟶𝐶))
19	16, 18	anbi12d 644	. . . 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)))
20	13, 19	bitrid 286	. . 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)))
21	12, 20	imbitrrid 249	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))))
22		xp1st 8037	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) → (1st ‘𝑦) ∈ (𝐶 ↑m 𝐴))
23	22	adantl 487	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → (1st ‘𝑦) ∈ (𝐶 ↑m 𝐴))
24		elmapi 8876	. . . . . 6 ⊢ ((1st ‘𝑦) ∈ (𝐶 ↑m 𝐴) → (1st ‘𝑦):𝐴⟶𝐶)
25	23, 24	syl 18	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → (1st ‘𝑦):𝐴⟶𝐶)
26		xp2nd 8038	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) → (2nd ‘𝑦) ∈ (𝐶 ↑m 𝐵))
27	26	adantl 487	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → (2nd ‘𝑦) ∈ (𝐶 ↑m 𝐵))
28		elmapi 8876	. . . . . 6 ⊢ ((2nd ‘𝑦) ∈ (𝐶 ↑m 𝐵) → (2nd ‘𝑦):𝐵⟶𝐶)
29	27, 28	syl 18	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → (2nd ‘𝑦):𝐵⟶𝐶)
30		simplr 781	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → (𝐴 ∩ 𝐵) = ∅)
31	25, 29, 30	fun2d 6751	. . . 4 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → ((1st ‘𝑦) ∪ (2nd ‘𝑦)):(𝐴 ∪ 𝐵)⟶𝐶)
32	31	ex 418	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) → ((1st ‘𝑦) ∪ (2nd ‘𝑦)):(𝐴 ∪ 𝐵)⟶𝐶))
33	15, 17	unexd 7772	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ V)
34	14, 33	elmapd 8867	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ↔ ((1st ‘𝑦) ∪ (2nd ‘𝑦)):(𝐴 ∪ 𝐵)⟶𝐶))
35	32, 34	sylibrd 262	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∈ (𝐶 ↑m (𝐴 ∪ 𝐵))))
36		1st2nd2 8044	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
37	36	ad2antll 742	. . . . . 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)⟩)
38	25	adantrl 729	. . . . . . . 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)
39	29	adantrl 729	. . . . . . . 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)
40		res0 5975	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ↾ ∅) = ∅
41		res0 5975	. . . . . . . . . 10 ⊢ ((2nd ‘𝑦) ↾ ∅) = ∅
42	40, 41	eqtr4i 2786	. . . . . . . . 9 ⊢ ((1st ‘𝑦) ↾ ∅) = ((2nd ‘𝑦) ↾ ∅)
43		simplr 781	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → (𝐴 ∩ 𝐵) = ∅)
44	43	reseq2d 5971	. . . . . . . . 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) ↾ ∅))
45	43	reseq2d 5971	. . . . . . . . 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) ↾ ∅))
46	42, 44, 45	3eqtr4a 2821	. . . . . . . 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)))
47		fresaunres1 6760	. . . . . . . 8 ⊢ (((1st ‘𝑦):𝐴⟶𝐶 ∧ (2nd ‘𝑦):𝐵⟶𝐶 ∧ ((1st ‘𝑦) ↾ (𝐴 ∩ 𝐵)) = ((2nd ‘𝑦) ↾ (𝐴 ∩ 𝐵))) → (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴) = (1st ‘𝑦))
48	38, 39, 46, 47	syl3anc 1398	. . . . . . 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))
49		fresaunres2 6759	. . . . . . . 8 ⊢ (((1st ‘𝑦):𝐴⟶𝐶 ∧ (2nd ‘𝑦):𝐵⟶𝐶 ∧ ((1st ‘𝑦) ↾ (𝐴 ∩ 𝐵)) = ((2nd ‘𝑦) ↾ (𝐴 ∩ 𝐵))) → (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵) = (2nd ‘𝑦))
50	38, 39, 46, 49	syl3anc 1398	. . . . . . 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))
51	48, 50	opeq12d 4847	. . . . . 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)⟩)
52	37, 51	eqtr4d 2798	. . . . 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)⟩)
53		reseq1 5965	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → (𝑥 ↾ 𝐴) = (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴))
54		reseq1 5965	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → (𝑥 ↾ 𝐵) = (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵))
55	53, 54	opeq12d 4847	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ = ⟨(((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴), (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵)⟩)
56	55	eqeq2d 2771	. . . . 5 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ ↔ 𝑦 = ⟨(((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴), (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵)⟩))
57	52, 56	syl5ibrcom 250	. . . 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)⟩))
58		ffn 6714	. . . . . . . 8 ⊢ (𝑥:(𝐴 ∪ 𝐵)⟶𝐶 → 𝑥 Fn (𝐴 ∪ 𝐵))
59		fnresdm 6663	. . . . . . . 8 ⊢ (𝑥 Fn (𝐴 ∪ 𝐵) → (𝑥 ↾ (𝐴 ∪ 𝐵)) = 𝑥)
60	5, 58, 59	3syl 19	. . . . . . 7 ⊢ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → (𝑥 ↾ (𝐴 ∪ 𝐵)) = 𝑥)
61	60	ad2antrl 741	. . . . . 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)
62	61	eqcomd 2766	. . . . 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)))
63		vex 3454	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
64	63	resex 6020	. . . . . . . . 9 ⊢ (𝑥 ↾ 𝐴) ∈ V
65	63	resex 6020	. . . . . . . . 9 ⊢ (𝑥 ↾ 𝐵) ∈ V
66	64, 65	op1std 8015	. . . . . . . 8 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (1st ‘𝑦) = (𝑥 ↾ 𝐴))
67	64, 65	op2ndd 8016	. . . . . . . 8 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (2nd ‘𝑦) = (𝑥 ↾ 𝐵))
68	66, 67	uneq12d 4167	. . . . . . 7 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) = ((𝑥 ↾ 𝐴) ∪ (𝑥 ↾ 𝐵)))
69		resundi 5985	. . . . . . 7 ⊢ (𝑥 ↾ (𝐴 ∪ 𝐵)) = ((𝑥 ↾ 𝐴) ∪ (𝑥 ↾ 𝐵))
70	68, 69	eqtr4di 2813	. . . . . 6 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) = (𝑥 ↾ (𝐴 ∪ 𝐵)))
71	70	eqeq2d 2771	. . . . 5 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↔ 𝑥 = (𝑥 ↾ (𝐴 ∪ 𝐵))))
72	62, 71	syl5ibrcom 250	. . . 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))))
73	57, 72	impbid 215	. . 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)⟩))
74	73	ex 418	. 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)⟩)))
75	1, 4, 21, 35, 74	en3d 9023	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑m (𝐴 ∪ 𝐵)) ≈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ⊆ wss 3896   ∩ cin 4088   ∪ cun 4148  ∅c0 4277  ⟨cop 4834   class class class wbr 5102   × cxp 5649   ↾ cres 5653   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  (class class class)co 7424  1^st c1st 8003  2^nd c2nd 8004   ↑_m cmap 8854   ≈ cen 8977

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-map 8856  df-en 8981

This theorem is used by:  map2xp  9173  mapdom2  9174  mapdjuen  10266  ackbij1lem5  10308  hashmap  14587  mpct  46323