Theorem mapunen 9090

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 7392	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑m (𝐴 ∪ 𝐵)) ∈ V)
2		ovexd 7392	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑m 𝐴) ∈ V)
3		ovexd 7392	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑m 𝐵) ∈ V)
4	2, 3	xpexd 7685	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) ∈ V)
5		elmapi 8787	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → 𝑥:(𝐴 ∪ 𝐵)⟶𝐶)
6		ssun1 4132	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
7		fssres 6708	. . . . 5 ⊢ ((𝑥:(𝐴 ∪ 𝐵)⟶𝐶 ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐴):𝐴⟶𝐶)
8	5, 6, 7	sylancl 586	. . . 4 ⊢ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐴):𝐴⟶𝐶)
9		ssun2 4133	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
10		fssres 6708	. . . . 5 ⊢ ((𝑥:(𝐴 ∪ 𝐵)⟶𝐶 ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐵):𝐵⟶𝐶)
11	5, 9, 10	sylancl 586	. . . 4 ⊢ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐵):𝐵⟶𝐶)
12	8, 11	jca 512	. . 3 ⊢ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → ((𝑥 ↾ 𝐴):𝐴⟶𝐶 ∧ (𝑥 ↾ 𝐵):𝐵⟶𝐶))
13		opelxp 5669	. . . 4 ⊢ (⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) ↔ ((𝑥 ↾ 𝐴) ∈ (𝐶 ↑m 𝐴) ∧ (𝑥 ↾ 𝐵) ∈ (𝐶 ↑m 𝐵)))
14		simpl3 1193	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐶 ∈ 𝑋)
15		simpl1 1191	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ 𝑉)
16	14, 15	elmapd 8779	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ↾ 𝐴) ∈ (𝐶 ↑m 𝐴) ↔ (𝑥 ↾ 𝐴):𝐴⟶𝐶))
17		simpl2 1192	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ 𝑊)
18	14, 17	elmapd 8779	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ↾ 𝐵) ∈ (𝐶 ↑m 𝐵) ↔ (𝑥 ↾ 𝐵):𝐵⟶𝐶))
19	16, 18	anbi12d 631	. . . 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 282	. . 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	syl5ibr 245	. 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 7953	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) → (1st ‘𝑦) ∈ (𝐶 ↑m 𝐴))
23	22	adantl 482	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → (1st ‘𝑦) ∈ (𝐶 ↑m 𝐴))
24		elmapi 8787	. . . . . 6 ⊢ ((1st ‘𝑦) ∈ (𝐶 ↑m 𝐴) → (1st ‘𝑦):𝐴⟶𝐶)
25	23, 24	syl 17	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → (1st ‘𝑦):𝐴⟶𝐶)
26		xp2nd 7954	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) → (2nd ‘𝑦) ∈ (𝐶 ↑m 𝐵))
27	26	adantl 482	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → (2nd ‘𝑦) ∈ (𝐶 ↑m 𝐵))
28		elmapi 8787	. . . . . 6 ⊢ ((2nd ‘𝑦) ∈ (𝐶 ↑m 𝐵) → (2nd ‘𝑦):𝐵⟶𝐶)
29	27, 28	syl 17	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → (2nd ‘𝑦):𝐵⟶𝐶)
30		simplr 767	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵))) → (𝐴 ∩ 𝐵) = ∅)
31	25, 29, 30	fun2d 6706	. . . 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 413	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) → ((1st ‘𝑦) ∪ (2nd ‘𝑦)):(𝐴 ∪ 𝐵)⟶𝐶))
33		unexg 7683	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
34	15, 17, 33	syl2anc 584	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ V)
35	14, 34	elmapd 8779	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ↔ ((1st ‘𝑦) ∪ (2nd ‘𝑦)):(𝐴 ∪ 𝐵)⟶𝐶))
36	32, 35	sylibrd 258	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ∈ (𝐶 ↑m (𝐴 ∪ 𝐵))))
37		1st2nd2 7960	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
38	37	ad2antll 727	. . . . . 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)⟩)
39	25	adantrl 714	. . . . . . . 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)
40	29	adantrl 714	. . . . . . . 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)
41		res0 5941	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ↾ ∅) = ∅
42		res0 5941	. . . . . . . . . 10 ⊢ ((2nd ‘𝑦) ↾ ∅) = ∅
43	41, 42	eqtr4i 2767	. . . . . . . . 9 ⊢ ((1st ‘𝑦) ↾ ∅) = ((2nd ‘𝑦) ↾ ∅)
44		simplr 767	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) ∧ 𝑦 ∈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))) → (𝐴 ∩ 𝐵) = ∅)
45	44	reseq2d 5937	. . . . . . . . 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) ↾ ∅))
46	44	reseq2d 5937	. . . . . . . . 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) ↾ ∅))
47	43, 45, 46	3eqtr4a 2802	. . . . . . . 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)))
48		fresaunres1 6715	. . . . . . . 8 ⊢ (((1st ‘𝑦):𝐴⟶𝐶 ∧ (2nd ‘𝑦):𝐵⟶𝐶 ∧ ((1st ‘𝑦) ↾ (𝐴 ∩ 𝐵)) = ((2nd ‘𝑦) ↾ (𝐴 ∩ 𝐵))) → (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴) = (1st ‘𝑦))
49	39, 40, 47, 48	syl3anc 1371	. . . . . . 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		fresaunres2 6714	. . . . . . . 8 ⊢ (((1st ‘𝑦):𝐴⟶𝐶 ∧ (2nd ‘𝑦):𝐵⟶𝐶 ∧ ((1st ‘𝑦) ↾ (𝐴 ∩ 𝐵)) = ((2nd ‘𝑦) ↾ (𝐴 ∩ 𝐵))) → (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵) = (2nd ‘𝑦))
51	39, 40, 47, 50	syl3anc 1371	. . . . . . 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 4838	. . . . . 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	38, 52	eqtr4d 2779	. . . . 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 5931	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → (𝑥 ↾ 𝐴) = (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴))
55		reseq1 5931	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → (𝑥 ↾ 𝐵) = (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵))
56	54, 55	opeq12d 4838	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ = ⟨(((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴), (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵)⟩)
57	56	eqeq2d 2747	. . . . 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 246	. . . 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 6668	. . . . . . . 8 ⊢ (𝑥:(𝐴 ∪ 𝐵)⟶𝐶 → 𝑥 Fn (𝐴 ∪ 𝐵))
60		fnresdm 6620	. . . . . . . 8 ⊢ (𝑥 Fn (𝐴 ∪ 𝐵) → (𝑥 ↾ (𝐴 ∪ 𝐵)) = 𝑥)
61	5, 59, 60	3syl 18	. . . . . . 7 ⊢ (𝑥 ∈ (𝐶 ↑m (𝐴 ∪ 𝐵)) → (𝑥 ↾ (𝐴 ∪ 𝐵)) = 𝑥)
62	61	ad2antrl 726	. . . . . 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 2742	. . . . 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 3449	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
65	64	resex 5985	. . . . . . . . 9 ⊢ (𝑥 ↾ 𝐴) ∈ V
66	64	resex 5985	. . . . . . . . 9 ⊢ (𝑥 ↾ 𝐵) ∈ V
67	65, 66	op1std 7931	. . . . . . . 8 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (1st ‘𝑦) = (𝑥 ↾ 𝐴))
68	65, 66	op2ndd 7932	. . . . . . . 8 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (2nd ‘𝑦) = (𝑥 ↾ 𝐵))
69	67, 68	uneq12d 4124	. . . . . . 7 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) = ((𝑥 ↾ 𝐴) ∪ (𝑥 ↾ 𝐵)))
70		resundi 5951	. . . . . . 7 ⊢ (𝑥 ↾ (𝐴 ∪ 𝐵)) = ((𝑥 ↾ 𝐴) ∪ (𝑥 ↾ 𝐵))
71	69, 70	eqtr4di 2794	. . . . . 6 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) = (𝑥 ↾ (𝐴 ∪ 𝐵)))
72	71	eqeq2d 2747	. . . . 5 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↔ 𝑥 = (𝑥 ↾ (𝐴 ∪ 𝐵))))
73	63, 72	syl5ibrcom 246	. . . 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 211	. . 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 413	. 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	1, 4, 21, 36, 75	en3d 8929	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑m (𝐴 ∪ 𝐵)) ≈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3445   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282  ⟨cop 4592   class class class wbr 5105   × cxp 5631   ↾ cres 5635   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  1^st c1st 7919  2^nd c2nd 7920   ↑_m cmap 8765   ≈ cen 8880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-en 8884

This theorem is referenced by:  map2xp  9091  mapdom2  9092  mapdjuen  10116  ackbij1lem5  10160  hashmap  14335  mpct  43412