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Theorem mapunen 9090
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 7392 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶m (𝐴𝐵)) ∈ V)
2 ovexd 7392 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶m 𝐴) ∈ V)
3 ovexd 7392 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶m 𝐵) ∈ V)
42, 3xpexd 7685 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝐶m 𝐴) × (𝐶m 𝐵)) ∈ V)
5 elmapi 8787 . . . . 5 (𝑥 ∈ (𝐶m (𝐴𝐵)) → 𝑥:(𝐴𝐵)⟶𝐶)
6 ssun1 4132 . . . . 5 𝐴 ⊆ (𝐴𝐵)
7 fssres 6708 . . . . 5 ((𝑥:(𝐴𝐵)⟶𝐶𝐴 ⊆ (𝐴𝐵)) → (𝑥𝐴):𝐴𝐶)
85, 6, 7sylancl 586 . . . 4 (𝑥 ∈ (𝐶m (𝐴𝐵)) → (𝑥𝐴):𝐴𝐶)
9 ssun2 4133 . . . . 5 𝐵 ⊆ (𝐴𝐵)
10 fssres 6708 . . . . 5 ((𝑥:(𝐴𝐵)⟶𝐶𝐵 ⊆ (𝐴𝐵)) → (𝑥𝐵):𝐵𝐶)
115, 9, 10sylancl 586 . . . 4 (𝑥 ∈ (𝐶m (𝐴𝐵)) → (𝑥𝐵):𝐵𝐶)
128, 11jca 512 . . 3 (𝑥 ∈ (𝐶m (𝐴𝐵)) → ((𝑥𝐴):𝐴𝐶 ∧ (𝑥𝐵):𝐵𝐶))
13 opelxp 5669 . . . 4 (⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) ↔ ((𝑥𝐴) ∈ (𝐶m 𝐴) ∧ (𝑥𝐵) ∈ (𝐶m 𝐵)))
14 simpl3 1193 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐶𝑋)
15 simpl1 1191 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐴𝑉)
1614, 15elmapd 8779 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥𝐴) ∈ (𝐶m 𝐴) ↔ (𝑥𝐴):𝐴𝐶))
17 simpl2 1192 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐵𝑊)
1814, 17elmapd 8779 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥𝐵) ∈ (𝐶m 𝐵) ↔ (𝑥𝐵):𝐵𝐶))
1916, 18anbi12d 631 . . . 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, 20syl5ibr 245 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑥 ∈ (𝐶m (𝐴𝐵)) → ⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))))
22 xp1st 7953 . . . . . . 7 (𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) → (1st𝑦) ∈ (𝐶m 𝐴))
2322adantl 482 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (1st𝑦) ∈ (𝐶m 𝐴))
24 elmapi 8787 . . . . . 6 ((1st𝑦) ∈ (𝐶m 𝐴) → (1st𝑦):𝐴𝐶)
2523, 24syl 17 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (1st𝑦):𝐴𝐶)
26 xp2nd 7954 . . . . . . 7 (𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) → (2nd𝑦) ∈ (𝐶m 𝐵))
2726adantl 482 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (2nd𝑦) ∈ (𝐶m 𝐵))
28 elmapi 8787 . . . . . 6 ((2nd𝑦) ∈ (𝐶m 𝐵) → (2nd𝑦):𝐵𝐶)
2927, 28syl 17 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (2nd𝑦):𝐵𝐶)
30 simplr 767 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (𝐴𝐵) = ∅)
3125, 29, 30fun2d 6706 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶)
3231ex 413 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) → ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶))
33 unexg 7683 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
3415, 17, 33syl2anc 584 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ∈ V)
3514, 34elmapd 8779 . . 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 7960 . . . . . . 7 (𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
3837ad2antll 727 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
3925adantrl 714 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (1st𝑦):𝐴𝐶)
4029adantrl 714 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (2nd𝑦):𝐵𝐶)
41 res0 5941 . . . . . . . . . 10 ((1st𝑦) ↾ ∅) = ∅
42 res0 5941 . . . . . . . . . 10 ((2nd𝑦) ↾ ∅) = ∅
4341, 42eqtr4i 2767 . . . . . . . . 9 ((1st𝑦) ↾ ∅) = ((2nd𝑦) ↾ ∅)
44 simplr 767 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (𝐴𝐵) = ∅)
4544reseq2d 5937 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → ((1st𝑦) ↾ (𝐴𝐵)) = ((1st𝑦) ↾ ∅))
4644reseq2d 5937 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → ((2nd𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ ∅))
4743, 45, 463eqtr4a 2802 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵)))
48 fresaunres1 6715 . . . . . . . 8 (((1st𝑦):𝐴𝐶 ∧ (2nd𝑦):𝐵𝐶 ∧ ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴) = (1st𝑦))
4939, 40, 47, 48syl3anc 1371 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴) = (1st𝑦))
50 fresaunres2 6714 . . . . . . . 8 (((1st𝑦):𝐴𝐶 ∧ (2nd𝑦):𝐵𝐶 ∧ ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵) = (2nd𝑦))
5139, 40, 47, 50syl3anc 1371 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵) = (2nd𝑦))
5249, 51opeq12d 4838 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
5338, 52eqtr4d 2779 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → 𝑦 = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩)
54 reseq1 5931 . . . . . . 7 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑥𝐴) = (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴))
55 reseq1 5931 . . . . . . 7 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑥𝐵) = (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵))
5654, 55opeq12d 4838 . . . . . 6 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → ⟨(𝑥𝐴), (𝑥𝐵)⟩ = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩)
5756eqeq2d 2747 . . . . 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 6668 . . . . . . . 8 (𝑥:(𝐴𝐵)⟶𝐶𝑥 Fn (𝐴𝐵))
60 fnresdm 6620 . . . . . . . 8 (𝑥 Fn (𝐴𝐵) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
615, 59, 603syl 18 . . . . . . 7 (𝑥 ∈ (𝐶m (𝐴𝐵)) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
6261ad2antrl 726 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
6362eqcomd 2742 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → 𝑥 = (𝑥 ↾ (𝐴𝐵)))
64 vex 3449 . . . . . . . . . 10 𝑥 ∈ V
6564resex 5985 . . . . . . . . 9 (𝑥𝐴) ∈ V
6664resex 5985 . . . . . . . . 9 (𝑥𝐵) ∈ V
6765, 66op1std 7931 . . . . . . . 8 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (1st𝑦) = (𝑥𝐴))
6865, 66op2ndd 7932 . . . . . . . 8 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (2nd𝑦) = (𝑥𝐵))
6967, 68uneq12d 4124 . . . . . . 7 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → ((1st𝑦) ∪ (2nd𝑦)) = ((𝑥𝐴) ∪ (𝑥𝐵)))
70 resundi 5951 . . . . . . 7 (𝑥 ↾ (𝐴𝐵)) = ((𝑥𝐴) ∪ (𝑥𝐵))
7169, 70eqtr4di 2794 . . . . . 6 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → ((1st𝑦) ∪ (2nd𝑦)) = (𝑥 ↾ (𝐴𝐵)))
7271eqeq2d 2747 . . . . 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 413 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) ↔ 𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩)))
761, 4, 21, 36, 75en3d 8929 1 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶m (𝐴𝐵)) ≈ ((𝐶m 𝐴) × (𝐶m 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  cun 3908  cin 3909  wss 3910  c0 4282  cop 4592   class class class wbr 5105   × cxp 5631  cres 5635   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  m cmap 8765  cen 8880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-en 8884
This theorem is referenced by:  map2xp  9091  mapdom2  9092  mapdjuen  10116  ackbij1lem5  10160  hashmap  14335  mpct  43412
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