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Theorem mapunen 8533
 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 7048 . . 3 (𝐶𝑚 (𝐴𝐵)) ∈ V
21a1i 11 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶𝑚 (𝐴𝐵)) ∈ V)
3 ovex 7048 . . . 4 (𝐶𝑚 𝐴) ∈ V
4 ovex 7048 . . . 4 (𝐶𝑚 𝐵) ∈ V
53, 4xpex 7333 . . 3 ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) ∈ V
65a1i 11 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) ∈ V)
7 elmapi 8278 . . . . 5 (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → 𝑥:(𝐴𝐵)⟶𝐶)
8 ssun1 4069 . . . . 5 𝐴 ⊆ (𝐴𝐵)
9 fssres 6412 . . . . 5 ((𝑥:(𝐴𝐵)⟶𝐶𝐴 ⊆ (𝐴𝐵)) → (𝑥𝐴):𝐴𝐶)
107, 8, 9sylancl 586 . . . 4 (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → (𝑥𝐴):𝐴𝐶)
11 ssun2 4070 . . . . 5 𝐵 ⊆ (𝐴𝐵)
12 fssres 6412 . . . . 5 ((𝑥:(𝐴𝐵)⟶𝐶𝐵 ⊆ (𝐴𝐵)) → (𝑥𝐵):𝐵𝐶)
137, 11, 12sylancl 586 . . . 4 (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → (𝑥𝐵):𝐵𝐶)
1410, 13jca 512 . . 3 (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → ((𝑥𝐴):𝐴𝐶 ∧ (𝑥𝐵):𝐵𝐶))
15 opelxp 5479 . . . 4 (⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) ↔ ((𝑥𝐴) ∈ (𝐶𝑚 𝐴) ∧ (𝑥𝐵) ∈ (𝐶𝑚 𝐵)))
16 simpl3 1186 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐶𝑋)
17 simpl1 1184 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐴𝑉)
1816, 17elmapd 8270 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥𝐴) ∈ (𝐶𝑚 𝐴) ↔ (𝑥𝐴):𝐴𝐶))
19 simpl2 1185 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐵𝑊)
2016, 19elmapd 8270 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥𝐵) ∈ (𝐶𝑚 𝐵) ↔ (𝑥𝐵):𝐵𝐶))
2118, 20anbi12d 630 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (((𝑥𝐴) ∈ (𝐶𝑚 𝐴) ∧ (𝑥𝐵) ∈ (𝐶𝑚 𝐵)) ↔ ((𝑥𝐴):𝐴𝐶 ∧ (𝑥𝐵):𝐵𝐶)))
2215, 21syl5bb 284 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) ↔ ((𝑥𝐴):𝐴𝐶 ∧ (𝑥𝐵):𝐵𝐶)))
2314, 22syl5ibr 247 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → ⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))))
24 xp1st 7577 . . . . . . 7 (𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) → (1st𝑦) ∈ (𝐶𝑚 𝐴))
2524adantl 482 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (1st𝑦) ∈ (𝐶𝑚 𝐴))
26 elmapi 8278 . . . . . 6 ((1st𝑦) ∈ (𝐶𝑚 𝐴) → (1st𝑦):𝐴𝐶)
2725, 26syl 17 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (1st𝑦):𝐴𝐶)
28 xp2nd 7578 . . . . . . 7 (𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) → (2nd𝑦) ∈ (𝐶𝑚 𝐵))
2928adantl 482 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (2nd𝑦) ∈ (𝐶𝑚 𝐵))
30 elmapi 8278 . . . . . 6 ((2nd𝑦) ∈ (𝐶𝑚 𝐵) → (2nd𝑦):𝐵𝐶)
3129, 30syl 17 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (2nd𝑦):𝐵𝐶)
32 simplr 765 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (𝐴𝐵) = ∅)
3327, 31, 32fun2d 6410 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶)
3433ex 413 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) → ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶))
35 unexg 7329 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
3617, 19, 35syl2anc 584 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ∈ V)
3716, 36elmapd 8270 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (((1st𝑦) ∪ (2nd𝑦)) ∈ (𝐶𝑚 (𝐴𝐵)) ↔ ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶))
3834, 37sylibrd 260 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) → ((1st𝑦) ∪ (2nd𝑦)) ∈ (𝐶𝑚 (𝐴𝐵))))
39 1st2nd2 7584 . . . . . . 7 (𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
4039ad2antll 725 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
4127adantrl 712 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (1st𝑦):𝐴𝐶)
4231adantrl 712 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (2nd𝑦):𝐵𝐶)
43 res0 5738 . . . . . . . . . 10 ((1st𝑦) ↾ ∅) = ∅
44 res0 5738 . . . . . . . . . 10 ((2nd𝑦) ↾ ∅) = ∅
4543, 44eqtr4i 2822 . . . . . . . . 9 ((1st𝑦) ↾ ∅) = ((2nd𝑦) ↾ ∅)
46 simplr 765 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (𝐴𝐵) = ∅)
4746reseq2d 5734 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → ((1st𝑦) ↾ (𝐴𝐵)) = ((1st𝑦) ↾ ∅))
4846reseq2d 5734 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → ((2nd𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ ∅))
4945, 47, 483eqtr4a 2857 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵)))
50 fresaunres1 6419 . . . . . . . 8 (((1st𝑦):𝐴𝐶 ∧ (2nd𝑦):𝐵𝐶 ∧ ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴) = (1st𝑦))
5141, 42, 49, 50syl3anc 1364 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴) = (1st𝑦))
52 fresaunres2 6418 . . . . . . . 8 (((1st𝑦):𝐴𝐶 ∧ (2nd𝑦):𝐵𝐶 ∧ ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵) = (2nd𝑦))
5341, 42, 49, 52syl3anc 1364 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵) = (2nd𝑦))
5451, 53opeq12d 4718 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
5540, 54eqtr4d 2834 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → 𝑦 = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩)
56 reseq1 5728 . . . . . . 7 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑥𝐴) = (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴))
57 reseq1 5728 . . . . . . 7 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑥𝐵) = (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵))
5856, 57opeq12d 4718 . . . . . 6 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → ⟨(𝑥𝐴), (𝑥𝐵)⟩ = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩)
5958eqeq2d 2805 . . . . 5 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ ↔ 𝑦 = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩))
6055, 59syl5ibrcom 248 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → 𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩))
61 ffn 6382 . . . . . . . 8 (𝑥:(𝐴𝐵)⟶𝐶𝑥 Fn (𝐴𝐵))
62 fnresdm 6336 . . . . . . . 8 (𝑥 Fn (𝐴𝐵) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
637, 61, 623syl 18 . . . . . . 7 (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
6463ad2antrl 724 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
6564eqcomd 2801 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → 𝑥 = (𝑥 ↾ (𝐴𝐵)))
66 vex 3440 . . . . . . . . . 10 𝑥 ∈ V
6766resex 5780 . . . . . . . . 9 (𝑥𝐴) ∈ V
6866resex 5780 . . . . . . . . 9 (𝑥𝐵) ∈ V
6967, 68op1std 7555 . . . . . . . 8 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (1st𝑦) = (𝑥𝐴))
7067, 68op2ndd 7556 . . . . . . . 8 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (2nd𝑦) = (𝑥𝐵))
7169, 70uneq12d 4061 . . . . . . 7 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → ((1st𝑦) ∪ (2nd𝑦)) = ((𝑥𝐴) ∪ (𝑥𝐵)))
72 resundi 5748 . . . . . . 7 (𝑥 ↾ (𝐴𝐵)) = ((𝑥𝐴) ∪ (𝑥𝐵))
7371, 72syl6eqr 2849 . . . . . 6 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → ((1st𝑦) ∪ (2nd𝑦)) = (𝑥 ↾ (𝐴𝐵)))
7473eqeq2d 2805 . . . . 5 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) ↔ 𝑥 = (𝑥 ↾ (𝐴𝐵))))
7565, 74syl5ibrcom 248 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → 𝑥 = ((1st𝑦) ∪ (2nd𝑦))))
7660, 75impbid 213 . . 3 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))) → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) ↔ 𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩))
7776ex 413 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥 ∈ (𝐶𝑚 (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵))) → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) ↔ 𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩)))
782, 6, 23, 38, 77en3d 8394 1 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶𝑚 (𝐴𝐵)) ≈ ((𝐶𝑚 𝐴) × (𝐶𝑚 𝐵)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080   = wceq 1522   ∈ wcel 2081  Vcvv 3437   ∪ cun 3857   ∩ cin 3858   ⊆ wss 3859  ∅c0 4211  ⟨cop 4478   class class class wbr 4962   × cxp 5441   ↾ cres 5445   Fn wfn 6220  ⟶wf 6221  ‘cfv 6225  (class class class)co 7016  1st c1st 7543  2nd c2nd 7544   ↑𝑚 cmap 8256   ≈ cen 8354 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-map 8258  df-en 8358 This theorem is referenced by:  map2xp  8534  mapdom2  8535  mapdjuen  9452  ackbij1lem5  9492  hashmap  13644  mpct  41004
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