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Theorem mapunen 8338
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 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶𝑚 (𝐴𝐵)) ≈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))

Proof of Theorem mapunen
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6876 . . 3 (𝐶𝑚 (𝐴𝐵)) ∈ V
21a1i 11 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶𝑚 (𝐴𝐵)) ∈ V)
3 ovex 6876 . . . 4 (𝐶𝑚 𝐴) ∈ V
4 ovex 6876 . . . 4 (𝐶𝑚 𝐵) ∈ V
53, 4xpex 7162 . . 3 ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) ∈ V
65a1i 11 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) ∈ V)
7 elmapi 8084 . . . . 5 (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → 𝑥:(𝐴𝐵)⟶𝐶)
8 ssun1 3940 . . . . 5 𝐴 ⊆ (𝐴𝐵)
9 fssres 6254 . . . . 5 ((𝑥:(𝐴𝐵)⟶𝐶𝐴 ⊆ (𝐴𝐵)) → (𝑥𝐴):𝐴𝐶)
107, 8, 9sylancl 580 . . . 4 (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → (𝑥𝐴):𝐴𝐶)
11 ssun2 3941 . . . . 5 𝐵 ⊆ (𝐴𝐵)
12 fssres 6254 . . . . 5 ((𝑥:(𝐴𝐵)⟶𝐶𝐵 ⊆ (𝐴𝐵)) → (𝑥𝐵):𝐵𝐶)
137, 11, 12sylancl 580 . . . 4 (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → (𝑥𝐵):𝐵𝐶)
1410, 13jca 507 . . 3 (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → ((𝑥𝐴):𝐴𝐶 ∧ (𝑥𝐵):𝐵𝐶))
15 opelxp 5315 . . . 4 (⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) ↔ ((𝑥𝐴) ∈ (𝐶𝑚 𝐴) ∧ (𝑥𝐵) ∈ (𝐶𝑚 𝐵)))
16 simpl3 1246 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐶𝑋)
17 simpl1 1242 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐴𝑉)
1816, 17elmapd 8076 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥𝐴) ∈ (𝐶𝑚 𝐴) ↔ (𝑥𝐴):𝐴𝐶))
19 simpl2 1244 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐵𝑊)
2016, 19elmapd 8076 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥𝐵) ∈ (𝐶𝑚 𝐵) ↔ (𝑥𝐵):𝐵𝐶))
2118, 20anbi12d 624 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (((𝑥𝐴) ∈ (𝐶𝑚 𝐴) ∧ (𝑥𝐵) ∈ (𝐶𝑚 𝐵)) ↔ ((𝑥𝐴):𝐴𝐶 ∧ (𝑥𝐵):𝐵𝐶)))
2215, 21syl5bb 274 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) ↔ ((𝑥𝐴):𝐴𝐶 ∧ (𝑥𝐵):𝐵𝐶)))
2314, 22syl5ibr 237 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → ⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))))
24 xp1st 7400 . . . . . . 7 (𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) → (1st𝑦) ∈ (𝐶𝑚 𝐴))
2524adantl 473 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (1st𝑦) ∈ (𝐶𝑚 𝐴))
26 elmapi 8084 . . . . . 6 ((1st𝑦) ∈ (𝐶𝑚 𝐴) → (1st𝑦):𝐴𝐶)
2725, 26syl 17 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (1st𝑦):𝐴𝐶)
28 xp2nd 7401 . . . . . . 7 (𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) → (2nd𝑦) ∈ (𝐶𝑚 𝐵))
2928adantl 473 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (2nd𝑦) ∈ (𝐶𝑚 𝐵))
30 elmapi 8084 . . . . . 6 ((2nd𝑦) ∈ (𝐶𝑚 𝐵) → (2nd𝑦):𝐵𝐶)
3129, 30syl 17 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (2nd𝑦):𝐵𝐶)
32 simplr 785 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (𝐴𝐵) = ∅)
33 fun2 6251 . . . . 5 ((((1st𝑦):𝐴𝐶 ∧ (2nd𝑦):𝐵𝐶) ∧ (𝐴𝐵) = ∅) → ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶)
3427, 31, 32, 33syl21anc 866 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶)
3534ex 401 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) → ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶))
36 unexg 7159 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
3717, 19, 36syl2anc 579 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ∈ V)
3816, 37elmapd 8076 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (((1st𝑦) ∪ (2nd𝑦)) ∈ (𝐶𝑚 (𝐴𝐵)) ↔ ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶))
3935, 38sylibrd 250 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) → ((1st𝑦) ∪ (2nd𝑦)) ∈ (𝐶𝑚 (𝐴𝐵))))
40 1st2nd2 7407 . . . . . . 7 (𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
4140ad2antll 720 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
4227adantrl 707 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (1st𝑦):𝐴𝐶)
4331adantrl 707 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (2nd𝑦):𝐵𝐶)
44 res0 5571 . . . . . . . . . 10 ((1st𝑦) ↾ ∅) = ∅
45 res0 5571 . . . . . . . . . 10 ((2nd𝑦) ↾ ∅) = ∅
4644, 45eqtr4i 2790 . . . . . . . . 9 ((1st𝑦) ↾ ∅) = ((2nd𝑦) ↾ ∅)
47 simplr 785 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (𝐴𝐵) = ∅)
4847reseq2d 5567 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → ((1st𝑦) ↾ (𝐴𝐵)) = ((1st𝑦) ↾ ∅))
4947reseq2d 5567 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → ((2nd𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ ∅))
5046, 48, 493eqtr4a 2825 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵)))
51 fresaunres1 6261 . . . . . . . 8 (((1st𝑦):𝐴𝐶 ∧ (2nd𝑦):𝐵𝐶 ∧ ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴) = (1st𝑦))
5242, 43, 50, 51syl3anc 1490 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴) = (1st𝑦))
53 fresaunres2 6260 . . . . . . . 8 (((1st𝑦):𝐴𝐶 ∧ (2nd𝑦):𝐵𝐶 ∧ ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵) = (2nd𝑦))
5442, 43, 50, 53syl3anc 1490 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵) = (2nd𝑦))
5552, 54opeq12d 4569 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
5641, 55eqtr4d 2802 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → 𝑦 = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩)
57 reseq1 5561 . . . . . . 7 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑥𝐴) = (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴))
58 reseq1 5561 . . . . . . 7 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑥𝐵) = (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵))
5957, 58opeq12d 4569 . . . . . 6 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → ⟨(𝑥𝐴), (𝑥𝐵)⟩ = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩)
6059eqeq2d 2775 . . . . 5 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ ↔ 𝑦 = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩))
6156, 60syl5ibrcom 238 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → 𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩))
62 ffn 6225 . . . . . . . 8 (𝑥:(𝐴𝐵)⟶𝐶𝑥 Fn (𝐴𝐵))
63 fnresdm 6180 . . . . . . . 8 (𝑥 Fn (𝐴𝐵) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
647, 62, 633syl 18 . . . . . . 7 (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
6564ad2antrl 719 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
6665eqcomd 2771 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → 𝑥 = (𝑥 ↾ (𝐴𝐵)))
67 vex 3353 . . . . . . . . . 10 𝑥 ∈ V
6867resex 5622 . . . . . . . . 9 (𝑥𝐴) ∈ V
6967resex 5622 . . . . . . . . 9 (𝑥𝐵) ∈ V
7068, 69op1std 7378 . . . . . . . 8 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (1st𝑦) = (𝑥𝐴))
7168, 69op2ndd 7379 . . . . . . . 8 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (2nd𝑦) = (𝑥𝐵))
7270, 71uneq12d 3932 . . . . . . 7 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → ((1st𝑦) ∪ (2nd𝑦)) = ((𝑥𝐴) ∪ (𝑥𝐵)))
73 resundi 5588 . . . . . . 7 (𝑥 ↾ (𝐴𝐵)) = ((𝑥𝐴) ∪ (𝑥𝐵))
7472, 73syl6eqr 2817 . . . . . 6 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → ((1st𝑦) ∪ (2nd𝑦)) = (𝑥 ↾ (𝐴𝐵)))
7574eqeq2d 2775 . . . . 5 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) ↔ 𝑥 = (𝑥 ↾ (𝐴𝐵))))
7666, 75syl5ibrcom 238 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → 𝑥 = ((1st𝑦) ∪ (2nd𝑦))))
7761, 76impbid 203 . . 3 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) ↔ 𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩))
7877ex 401 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) ↔ 𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩)))
792, 6, 23, 39, 78en3d 8199 1 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶𝑚 (𝐴𝐵)) ≈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  Vcvv 3350  cun 3732  cin 3733  wss 3734  c0 4081  cop 4342   class class class wbr 4811   × cxp 5277  cres 5281   Fn wfn 6065  wf 6066  cfv 6070  (class class class)co 6844  1st c1st 7366  2nd c2nd 7367  𝑚 cmap 8062  cen 8159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-1st 7368  df-2nd 7369  df-map 8064  df-en 8163
This theorem is referenced by:  map2xp  8339  mapdom2  8340  mapcdaen  9261  ackbij1lem5  9301  hashmap  13426  mpct  40041
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