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Theorem mapunen 9148
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 ovexd 7446 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶m (𝐴𝐵)) ∈ V)
2 ovexd 7446 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶m 𝐴) ∈ V)
3 ovexd 7446 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶m 𝐵) ∈ V)
42, 3xpexd 7740 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝐶m 𝐴) × (𝐶m 𝐵)) ∈ V)
5 elmapi 8845 . . . . 5 (𝑥 ∈ (𝐶m (𝐴𝐵)) → 𝑥:(𝐴𝐵)⟶𝐶)
6 ssun1 4171 . . . . 5 𝐴 ⊆ (𝐴𝐵)
7 fssres 6756 . . . . 5 ((𝑥:(𝐴𝐵)⟶𝐶𝐴 ⊆ (𝐴𝐵)) → (𝑥𝐴):𝐴𝐶)
85, 6, 7sylancl 584 . . . 4 (𝑥 ∈ (𝐶m (𝐴𝐵)) → (𝑥𝐴):𝐴𝐶)
9 ssun2 4172 . . . . 5 𝐵 ⊆ (𝐴𝐵)
10 fssres 6756 . . . . 5 ((𝑥:(𝐴𝐵)⟶𝐶𝐵 ⊆ (𝐴𝐵)) → (𝑥𝐵):𝐵𝐶)
115, 9, 10sylancl 584 . . . 4 (𝑥 ∈ (𝐶m (𝐴𝐵)) → (𝑥𝐵):𝐵𝐶)
128, 11jca 510 . . 3 (𝑥 ∈ (𝐶m (𝐴𝐵)) → ((𝑥𝐴):𝐴𝐶 ∧ (𝑥𝐵):𝐵𝐶))
13 opelxp 5711 . . . 4 (⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) ↔ ((𝑥𝐴) ∈ (𝐶m 𝐴) ∧ (𝑥𝐵) ∈ (𝐶m 𝐵)))
14 simpl3 1191 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐶𝑋)
15 simpl1 1189 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐴𝑉)
1614, 15elmapd 8836 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥𝐴) ∈ (𝐶m 𝐴) ↔ (𝑥𝐴):𝐴𝐶))
17 simpl2 1190 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐵𝑊)
1814, 17elmapd 8836 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥𝐵) ∈ (𝐶m 𝐵) ↔ (𝑥𝐵):𝐵𝐶))
1916, 18anbi12d 629 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (((𝑥𝐴) ∈ (𝐶m 𝐴) ∧ (𝑥𝐵) ∈ (𝐶m 𝐵)) ↔ ((𝑥𝐴):𝐴𝐶 ∧ (𝑥𝐵):𝐵𝐶)))
2013, 19bitrid 282 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) ↔ ((𝑥𝐴):𝐴𝐶 ∧ (𝑥𝐵):𝐵𝐶)))
2112, 20imbitrrid 245 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑥 ∈ (𝐶m (𝐴𝐵)) → ⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))))
22 xp1st 8009 . . . . . . 7 (𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) → (1st𝑦) ∈ (𝐶m 𝐴))
2322adantl 480 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (1st𝑦) ∈ (𝐶m 𝐴))
24 elmapi 8845 . . . . . 6 ((1st𝑦) ∈ (𝐶m 𝐴) → (1st𝑦):𝐴𝐶)
2523, 24syl 17 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (1st𝑦):𝐴𝐶)
26 xp2nd 8010 . . . . . . 7 (𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) → (2nd𝑦) ∈ (𝐶m 𝐵))
2726adantl 480 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (2nd𝑦) ∈ (𝐶m 𝐵))
28 elmapi 8845 . . . . . 6 ((2nd𝑦) ∈ (𝐶m 𝐵) → (2nd𝑦):𝐵𝐶)
2927, 28syl 17 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (2nd𝑦):𝐵𝐶)
30 simplr 765 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (𝐴𝐵) = ∅)
3125, 29, 30fun2d 6754 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶)
3231ex 411 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) → ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶))
33 unexg 7738 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
3415, 17, 33syl2anc 582 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ∈ V)
3514, 34elmapd 8836 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (((1st𝑦) ∪ (2nd𝑦)) ∈ (𝐶m (𝐴𝐵)) ↔ ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶))
3632, 35sylibrd 258 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) → ((1st𝑦) ∪ (2nd𝑦)) ∈ (𝐶m (𝐴𝐵))))
37 1st2nd2 8016 . . . . . . 7 (𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
3837ad2antll 725 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
3925adantrl 712 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (1st𝑦):𝐴𝐶)
4029adantrl 712 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (2nd𝑦):𝐵𝐶)
41 res0 5984 . . . . . . . . . 10 ((1st𝑦) ↾ ∅) = ∅
42 res0 5984 . . . . . . . . . 10 ((2nd𝑦) ↾ ∅) = ∅
4341, 42eqtr4i 2761 . . . . . . . . 9 ((1st𝑦) ↾ ∅) = ((2nd𝑦) ↾ ∅)
44 simplr 765 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (𝐴𝐵) = ∅)
4544reseq2d 5980 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → ((1st𝑦) ↾ (𝐴𝐵)) = ((1st𝑦) ↾ ∅))
4644reseq2d 5980 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → ((2nd𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ ∅))
4743, 45, 463eqtr4a 2796 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵)))
48 fresaunres1 6763 . . . . . . . 8 (((1st𝑦):𝐴𝐶 ∧ (2nd𝑦):𝐵𝐶 ∧ ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴) = (1st𝑦))
4939, 40, 47, 48syl3anc 1369 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴) = (1st𝑦))
50 fresaunres2 6762 . . . . . . . 8 (((1st𝑦):𝐴𝐶 ∧ (2nd𝑦):𝐵𝐶 ∧ ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵) = (2nd𝑦))
5139, 40, 47, 50syl3anc 1369 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵) = (2nd𝑦))
5249, 51opeq12d 4880 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
5338, 52eqtr4d 2773 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → 𝑦 = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩)
54 reseq1 5974 . . . . . . 7 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑥𝐴) = (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴))
55 reseq1 5974 . . . . . . 7 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑥𝐵) = (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵))
5654, 55opeq12d 4880 . . . . . 6 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → ⟨(𝑥𝐴), (𝑥𝐵)⟩ = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩)
5756eqeq2d 2741 . . . . 5 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ ↔ 𝑦 = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩))
5853, 57syl5ibrcom 246 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → 𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩))
59 ffn 6716 . . . . . . . 8 (𝑥:(𝐴𝐵)⟶𝐶𝑥 Fn (𝐴𝐵))
60 fnresdm 6668 . . . . . . . 8 (𝑥 Fn (𝐴𝐵) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
615, 59, 603syl 18 . . . . . . 7 (𝑥 ∈ (𝐶m (𝐴𝐵)) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
6261ad2antrl 724 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
6362eqcomd 2736 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → 𝑥 = (𝑥 ↾ (𝐴𝐵)))
64 vex 3476 . . . . . . . . . 10 𝑥 ∈ V
6564resex 6028 . . . . . . . . 9 (𝑥𝐴) ∈ V
6664resex 6028 . . . . . . . . 9 (𝑥𝐵) ∈ V
6765, 66op1std 7987 . . . . . . . 8 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (1st𝑦) = (𝑥𝐴))
6865, 66op2ndd 7988 . . . . . . . 8 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (2nd𝑦) = (𝑥𝐵))
6967, 68uneq12d 4163 . . . . . . 7 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → ((1st𝑦) ∪ (2nd𝑦)) = ((𝑥𝐴) ∪ (𝑥𝐵)))
70 resundi 5994 . . . . . . 7 (𝑥 ↾ (𝐴𝐵)) = ((𝑥𝐴) ∪ (𝑥𝐵))
7169, 70eqtr4di 2788 . . . . . 6 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → ((1st𝑦) ∪ (2nd𝑦)) = (𝑥 ↾ (𝐴𝐵)))
7271eqeq2d 2741 . . . . 5 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) ↔ 𝑥 = (𝑥 ↾ (𝐴𝐵))))
7363, 72syl5ibrcom 246 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → 𝑥 = ((1st𝑦) ∪ (2nd𝑦))))
7458, 73impbid 211 . . 3 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) ↔ 𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩))
7574ex 411 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) ↔ 𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩)))
761, 4, 21, 36, 75en3d 8987 1 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶m (𝐴𝐵)) ≈ ((𝐶m 𝐴) × (𝐶m 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1085   = wceq 1539  wcel 2104  Vcvv 3472  cun 3945  cin 3946  wss 3947  c0 4321  cop 4633   class class class wbr 5147   × cxp 5673  cres 5677   Fn wfn 6537  wf 6538  cfv 6542  (class class class)co 7411  1st c1st 7975  2nd c2nd 7976  m cmap 8822  cen 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824  df-en 8942
This theorem is referenced by:  map2xp  9149  mapdom2  9150  mapdjuen  10177  ackbij1lem5  10221  hashmap  14399  mpct  44198
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