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Theorem mapunen 8245
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 6793 . . 3 (𝐶𝑚 (𝐴𝐵)) ∈ V
21a1i 11 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶𝑚 (𝐴𝐵)) ∈ V)
3 ovex 6793 . . . 4 (𝐶𝑚 𝐴) ∈ V
4 ovex 6793 . . . 4 (𝐶𝑚 𝐵) ∈ V
53, 4xpex 7079 . . 3 ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) ∈ V
65a1i 11 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) ∈ V)
7 elmapi 7996 . . . . 5 (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → 𝑥:(𝐴𝐵)⟶𝐶)
8 ssun1 3884 . . . . 5 𝐴 ⊆ (𝐴𝐵)
9 fssres 6183 . . . . 5 ((𝑥:(𝐴𝐵)⟶𝐶𝐴 ⊆ (𝐴𝐵)) → (𝑥𝐴):𝐴𝐶)
107, 8, 9sylancl 697 . . . 4 (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → (𝑥𝐴):𝐴𝐶)
11 ssun2 3885 . . . . 5 𝐵 ⊆ (𝐴𝐵)
12 fssres 6183 . . . . 5 ((𝑥:(𝐴𝐵)⟶𝐶𝐵 ⊆ (𝐴𝐵)) → (𝑥𝐵):𝐵𝐶)
137, 11, 12sylancl 697 . . . 4 (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → (𝑥𝐵):𝐵𝐶)
1410, 13jca 555 . . 3 (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → ((𝑥𝐴):𝐴𝐶 ∧ (𝑥𝐵):𝐵𝐶))
15 opelxp 5255 . . . 4 (⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) ↔ ((𝑥𝐴) ∈ (𝐶𝑚 𝐴) ∧ (𝑥𝐵) ∈ (𝐶𝑚 𝐵)))
16 simpl3 1208 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐶𝑋)
17 simpl1 1204 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐴𝑉)
1816, 17elmapd 7988 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥𝐴) ∈ (𝐶𝑚 𝐴) ↔ (𝑥𝐴):𝐴𝐶))
19 simpl2 1206 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐵𝑊)
2016, 19elmapd 7988 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥𝐵) ∈ (𝐶𝑚 𝐵) ↔ (𝑥𝐵):𝐵𝐶))
2118, 20anbi12d 749 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (((𝑥𝐴) ∈ (𝐶𝑚 𝐴) ∧ (𝑥𝐵) ∈ (𝐶𝑚 𝐵)) ↔ ((𝑥𝐴):𝐴𝐶 ∧ (𝑥𝐵):𝐵𝐶)))
2215, 21syl5bb 272 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) ↔ ((𝑥𝐴):𝐴𝐶 ∧ (𝑥𝐵):𝐵𝐶)))
2314, 22syl5ibr 236 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → ⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))))
24 xp1st 7317 . . . . . . 7 (𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) → (1st𝑦) ∈ (𝐶𝑚 𝐴))
2524adantl 473 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (1st𝑦) ∈ (𝐶𝑚 𝐴))
26 elmapi 7996 . . . . . 6 ((1st𝑦) ∈ (𝐶𝑚 𝐴) → (1st𝑦):𝐴𝐶)
2725, 26syl 17 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (1st𝑦):𝐴𝐶)
28 xp2nd 7318 . . . . . . 7 (𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) → (2nd𝑦) ∈ (𝐶𝑚 𝐵))
2928adantl 473 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (2nd𝑦) ∈ (𝐶𝑚 𝐵))
30 elmapi 7996 . . . . . 6 ((2nd𝑦) ∈ (𝐶𝑚 𝐵) → (2nd𝑦):𝐵𝐶)
3129, 30syl 17 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (2nd𝑦):𝐵𝐶)
32 simplr 809 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (𝐴𝐵) = ∅)
33 fun2 6180 . . . . 5 ((((1st𝑦):𝐴𝐶 ∧ (2nd𝑦):𝐵𝐶) ∧ (𝐴𝐵) = ∅) → ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶)
3427, 31, 32, 33syl21anc 1438 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶)
3534ex 449 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) → ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶))
36 unexg 7076 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
3717, 19, 36syl2anc 696 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ∈ V)
3816, 37elmapd 7988 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (((1st𝑦) ∪ (2nd𝑦)) ∈ (𝐶𝑚 (𝐴𝐵)) ↔ ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶))
3935, 38sylibrd 249 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) → ((1st𝑦) ∪ (2nd𝑦)) ∈ (𝐶𝑚 (𝐴𝐵))))
40 1st2nd2 7324 . . . . . . 7 (𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
4140ad2antll 767 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
4227adantrl 754 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (1st𝑦):𝐴𝐶)
4331adantrl 754 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (2nd𝑦):𝐵𝐶)
44 res0 5507 . . . . . . . . . 10 ((1st𝑦) ↾ ∅) = ∅
45 res0 5507 . . . . . . . . . 10 ((2nd𝑦) ↾ ∅) = ∅
4644, 45eqtr4i 2749 . . . . . . . . 9 ((1st𝑦) ↾ ∅) = ((2nd𝑦) ↾ ∅)
47 simplr 809 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (𝐴𝐵) = ∅)
4847reseq2d 5503 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → ((1st𝑦) ↾ (𝐴𝐵)) = ((1st𝑦) ↾ ∅))
4947reseq2d 5503 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → ((2nd𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ ∅))
5046, 48, 493eqtr4a 2784 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵)))
51 fresaunres1 6190 . . . . . . . 8 (((1st𝑦):𝐴𝐶 ∧ (2nd𝑦):𝐵𝐶 ∧ ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴) = (1st𝑦))
5242, 43, 50, 51syl3anc 1439 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴) = (1st𝑦))
53 fresaunres2 6189 . . . . . . . 8 (((1st𝑦):𝐴𝐶 ∧ (2nd𝑦):𝐵𝐶 ∧ ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵) = (2nd𝑦))
5442, 43, 50, 53syl3anc 1439 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵) = (2nd𝑦))
5552, 54opeq12d 4517 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
5641, 55eqtr4d 2761 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → 𝑦 = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩)
57 reseq1 5497 . . . . . . 7 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑥𝐴) = (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴))
58 reseq1 5497 . . . . . . 7 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑥𝐵) = (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵))
5957, 58opeq12d 4517 . . . . . 6 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → ⟨(𝑥𝐴), (𝑥𝐵)⟩ = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩)
6059eqeq2d 2734 . . . . 5 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ ↔ 𝑦 = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩))
6156, 60syl5ibrcom 237 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → 𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩))
62 ffn 6158 . . . . . . . 8 (𝑥:(𝐴𝐵)⟶𝐶𝑥 Fn (𝐴𝐵))
63 fnresdm 6113 . . . . . . . 8 (𝑥 Fn (𝐴𝐵) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
647, 62, 633syl 18 . . . . . . 7 (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
6564ad2antrl 766 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
6665eqcomd 2730 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → 𝑥 = (𝑥 ↾ (𝐴𝐵)))
67 vex 3307 . . . . . . . . . 10 𝑥 ∈ V
6867resex 5553 . . . . . . . . 9 (𝑥𝐴) ∈ V
6967resex 5553 . . . . . . . . 9 (𝑥𝐵) ∈ V
7068, 69op1std 7295 . . . . . . . 8 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (1st𝑦) = (𝑥𝐴))
7168, 69op2ndd 7296 . . . . . . . 8 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (2nd𝑦) = (𝑥𝐵))
7270, 71uneq12d 3876 . . . . . . 7 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → ((1st𝑦) ∪ (2nd𝑦)) = ((𝑥𝐴) ∪ (𝑥𝐵)))
73 resundi 5520 . . . . . . 7 (𝑥 ↾ (𝐴𝐵)) = ((𝑥𝐴) ∪ (𝑥𝐵))
7472, 73syl6eqr 2776 . . . . . 6 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → ((1st𝑦) ∪ (2nd𝑦)) = (𝑥 ↾ (𝐴𝐵)))
7574eqeq2d 2734 . . . . 5 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) ↔ 𝑥 = (𝑥 ↾ (𝐴𝐵))))
7666, 75syl5ibrcom 237 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → 𝑥 = ((1st𝑦) ∪ (2nd𝑦))))
7761, 76impbid 202 . . 3 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) ↔ 𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩))
7877ex 449 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) ↔ 𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩)))
792, 6, 23, 39, 78en3d 8109 1 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶𝑚 (𝐴𝐵)) ≈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  Vcvv 3304  cun 3678  cin 3679  wss 3680  c0 4023  cop 4291   class class class wbr 4760   × cxp 5216  cres 5220   Fn wfn 5996  wf 5997  cfv 6001  (class class class)co 6765  1st c1st 7283  2nd c2nd 7284  𝑚 cmap 7974  cen 8069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-1st 7285  df-2nd 7286  df-map 7976  df-en 8073
This theorem is referenced by:  map2xp  8246  mapdom2  8247  mapcdaen  9119  ackbij1lem5  9159  hashmap  13335  mpct  39809
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