Theorem mapunen 8014

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 6577	. . 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 6577	. . . 4 ⊢ (𝐶 ↑𝑚 𝐴) ∈ V
4		ovex 6577	. . . 4 ⊢ (𝐶 ↑𝑚 𝐵) ∈ V
5	3, 4	xpex 6860	. . 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 7765	. . . . 5 ⊢ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → 𝑥:(𝐴 ∪ 𝐵)⟶𝐶)
8		ssun1 3738	. . . . 5 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
9		fssres 5983	. . . . 5 ⊢ ((𝑥:(𝐴 ∪ 𝐵)⟶𝐶 ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐴):𝐴⟶𝐶)
10	7, 8, 9	sylancl 693	. . . 4 ⊢ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐴):𝐴⟶𝐶)
11		ssun2 3739	. . . . 5 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
12		fssres 5983	. . . . 5 ⊢ ((𝑥:(𝐴 ∪ 𝐵)⟶𝐶 ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐵):𝐵⟶𝐶)
13	7, 11, 12	sylancl 693	. . . 4 ⊢ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → (𝑥 ↾ 𝐵):𝐵⟶𝐶)
14	10, 13	jca 553	. . 3 ⊢ (𝑥 ∈ (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) → ((𝑥 ↾ 𝐴):𝐴⟶𝐶 ∧ (𝑥 ↾ 𝐵):𝐵⟶𝐶))
15		opelxp 5070	. . . 4 ⊢ (⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) ↔ ((𝑥 ↾ 𝐴) ∈ (𝐶 ↑𝑚 𝐴) ∧ (𝑥 ↾ 𝐵) ∈ (𝐶 ↑𝑚 𝐵)))
16		simpl3 1059	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐶 ∈ 𝑋)
17		simpl1 1057	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ 𝑉)
18	16, 17	elmapd 7758	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ↾ 𝐴) ∈ (𝐶 ↑𝑚 𝐴) ↔ (𝑥 ↾ 𝐴):𝐴⟶𝐶))
19		simpl2 1058	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ 𝑊)
20	16, 19	elmapd 7758	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ↾ 𝐵) ∈ (𝐶 ↑𝑚 𝐵) ↔ (𝑥 ↾ 𝐵):𝐵⟶𝐶))
21	18, 20	anbi12d 743	. . . 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 271	. . 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 235	. 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 7089	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) → (1st ‘𝑦) ∈ (𝐶 ↑𝑚 𝐴))
25	24	adantl 481	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → (1st ‘𝑦) ∈ (𝐶 ↑𝑚 𝐴))
26		elmapi 7765	. . . . . 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 7090	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) → (2nd ‘𝑦) ∈ (𝐶 ↑𝑚 𝐵))
29	28	adantl 481	. . . . . 6 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → (2nd ‘𝑦) ∈ (𝐶 ↑𝑚 𝐵))
30		elmapi 7765	. . . . . 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 788	. . . . 5 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵))) → (𝐴 ∩ 𝐵) = ∅)
33		fun2 5980	. . . . 5 ⊢ ((((1st ‘𝑦):𝐴⟶𝐶 ∧ (2nd ‘𝑦):𝐵⟶𝐶) ∧ (𝐴 ∩ 𝐵) = ∅) → ((1st ‘𝑦) ∪ (2nd ‘𝑦)):(𝐴 ∪ 𝐵)⟶𝐶)
34	27, 31, 32, 33	syl21anc 1317	. . . 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 449	. . 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 6857	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∪ 𝐵) ∈ V)
37	17, 19, 36	syl2anc 691	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ V)
38	16, 37	elmapd 7758	. . 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 248	. 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 7096	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
41	40	ad2antll 761	. . . . . 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 748	. . . . . . . 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 748	. . . . . . . 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 5321	. . . . . . . . . 10 ⊢ ((1st ‘𝑦) ↾ ∅) = ∅
45		res0 5321	. . . . . . . . . 10 ⊢ ((2nd ‘𝑦) ↾ ∅) = ∅
46	44, 45	eqtr4i 2635	. . . . . . . . 9 ⊢ ((1st ‘𝑦) ↾ ∅) = ((2nd ‘𝑦) ↾ ∅)
47		simplr 788	. . . . . . . . . 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 5317	. . . . . . . . 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 5317	. . . . . . . . 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 2670	. . . . . . . 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 5990	. . . . . . . 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 1318	. . . . . . 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 5989	. . . . . . . 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 1318	. . . . . . 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 4348	. . . . . 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 2647	. . . . 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 5311	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → (𝑥 ↾ 𝐴) = (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴))
58		reseq1 5311	. . . . . . 7 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → (𝑥 ↾ 𝐵) = (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵))
59	57, 58	opeq12d 4348	. . . . . 6 ⊢ (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) → ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ = ⟨(((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐴), (((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↾ 𝐵)⟩)
60	59	eqeq2d 2620	. . . . 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 236	. . . 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 5958	. . . . . . . 8 ⊢ (𝑥:(𝐴 ∪ 𝐵)⟶𝐶 → 𝑥 Fn (𝐴 ∪ 𝐵))
63		fnresdm 5914	. . . . . . . 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 760	. . . . . 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 2616	. . . . 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 3176	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
68	67	resex 5363	. . . . . . . . 9 ⊢ (𝑥 ↾ 𝐴) ∈ V
69	67	resex 5363	. . . . . . . . 9 ⊢ (𝑥 ↾ 𝐵) ∈ V
70	68, 69	op1std 7069	. . . . . . . 8 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (1st ‘𝑦) = (𝑥 ↾ 𝐴))
71	68, 69	op2ndd 7070	. . . . . . . 8 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (2nd ‘𝑦) = (𝑥 ↾ 𝐵))
72	70, 71	uneq12d 3730	. . . . . . 7 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) = ((𝑥 ↾ 𝐴) ∪ (𝑥 ↾ 𝐵)))
73		resundi 5330	. . . . . . 7 ⊢ (𝑥 ↾ (𝐴 ∪ 𝐵)) = ((𝑥 ↾ 𝐴) ∪ (𝑥 ↾ 𝐵))
74	72, 73	syl6eqr 2662	. . . . . 6 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → ((1st ‘𝑦) ∪ (2nd ‘𝑦)) = (𝑥 ↾ (𝐴 ∪ 𝐵)))
75	74	eqeq2d 2620	. . . . 5 ⊢ (𝑦 = ⟨(𝑥 ↾ 𝐴), (𝑥 ↾ 𝐵)⟩ → (𝑥 = ((1st ‘𝑦) ∪ (2nd ‘𝑦)) ↔ 𝑥 = (𝑥 ↾ (𝐴 ∪ 𝐵))))
76	66, 75	syl5ibrcom 236	. . . 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 201	. . 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 449	. 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 7878	1 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑𝑚 (𝐴 ∪ 𝐵)) ≈ ((𝐶 ↑𝑚 𝐴) × (𝐶 ↑𝑚 𝐵)))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ⟨cop 4131   class class class wbr 4583   × cxp 5036   ↾ cres 5040   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058   ↑_𝑚 cmap 7744   ≈ cen 7838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-en 7842

This theorem is referenced by:  map2xp  8015  mapdom2  8016  mapcdaen  8889  ackbij1lem5  8929  hashmap  13082  mpct  38388