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Theorem mapunen 9185
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 7466 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶m (𝐴𝐵)) ∈ V)
2 ovexd 7466 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶m 𝐴) ∈ V)
3 ovexd 7466 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶m 𝐵) ∈ V)
42, 3xpexd 7770 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝐶m 𝐴) × (𝐶m 𝐵)) ∈ V)
5 elmapi 8888 . . . . 5 (𝑥 ∈ (𝐶m (𝐴𝐵)) → 𝑥:(𝐴𝐵)⟶𝐶)
6 ssun1 4188 . . . . 5 𝐴 ⊆ (𝐴𝐵)
7 fssres 6775 . . . . 5 ((𝑥:(𝐴𝐵)⟶𝐶𝐴 ⊆ (𝐴𝐵)) → (𝑥𝐴):𝐴𝐶)
85, 6, 7sylancl 586 . . . 4 (𝑥 ∈ (𝐶m (𝐴𝐵)) → (𝑥𝐴):𝐴𝐶)
9 ssun2 4189 . . . . 5 𝐵 ⊆ (𝐴𝐵)
10 fssres 6775 . . . . 5 ((𝑥:(𝐴𝐵)⟶𝐶𝐵 ⊆ (𝐴𝐵)) → (𝑥𝐵):𝐵𝐶)
115, 9, 10sylancl 586 . . . 4 (𝑥 ∈ (𝐶m (𝐴𝐵)) → (𝑥𝐵):𝐵𝐶)
128, 11jca 511 . . 3 (𝑥 ∈ (𝐶m (𝐴𝐵)) → ((𝑥𝐴):𝐴𝐶 ∧ (𝑥𝐵):𝐵𝐶))
13 opelxp 5725 . . . 4 (⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) ↔ ((𝑥𝐴) ∈ (𝐶m 𝐴) ∧ (𝑥𝐵) ∈ (𝐶m 𝐵)))
14 simpl3 1192 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐶𝑋)
15 simpl1 1190 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐴𝑉)
1614, 15elmapd 8879 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥𝐴) ∈ (𝐶m 𝐴) ↔ (𝑥𝐴):𝐴𝐶))
17 simpl2 1191 . . . . . 6 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → 𝐵𝑊)
1814, 17elmapd 8879 . . . . 5 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥𝐵) ∈ (𝐶m 𝐵) ↔ (𝑥𝐵):𝐵𝐶))
1916, 18anbi12d 632 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (((𝑥𝐴) ∈ (𝐶m 𝐴) ∧ (𝑥𝐵) ∈ (𝐶m 𝐵)) ↔ ((𝑥𝐴):𝐴𝐶 ∧ (𝑥𝐵):𝐵𝐶)))
2013, 19bitrid 283 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) ↔ ((𝑥𝐴):𝐴𝐶 ∧ (𝑥𝐵):𝐵𝐶)))
2112, 20imbitrrid 246 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑥 ∈ (𝐶m (𝐴𝐵)) → ⟨(𝑥𝐴), (𝑥𝐵)⟩ ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))))
22 xp1st 8045 . . . . . . 7 (𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) → (1st𝑦) ∈ (𝐶m 𝐴))
2322adantl 481 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (1st𝑦) ∈ (𝐶m 𝐴))
24 elmapi 8888 . . . . . 6 ((1st𝑦) ∈ (𝐶m 𝐴) → (1st𝑦):𝐴𝐶)
2523, 24syl 17 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (1st𝑦):𝐴𝐶)
26 xp2nd 8046 . . . . . . 7 (𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) → (2nd𝑦) ∈ (𝐶m 𝐵))
2726adantl 481 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (2nd𝑦) ∈ (𝐶m 𝐵))
28 elmapi 8888 . . . . . 6 ((2nd𝑦) ∈ (𝐶m 𝐵) → (2nd𝑦):𝐵𝐶)
2927, 28syl 17 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (2nd𝑦):𝐵𝐶)
30 simplr 769 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (𝐴𝐵) = ∅)
3125, 29, 30fun2d 6773 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶)
3231ex 412 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) → ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶))
33 unexg 7762 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ∈ V)
3415, 17, 33syl2anc 584 . . . 4 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ∈ V)
3514, 34elmapd 8879 . . 3 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (((1st𝑦) ∪ (2nd𝑦)) ∈ (𝐶m (𝐴𝐵)) ↔ ((1st𝑦) ∪ (2nd𝑦)):(𝐴𝐵)⟶𝐶))
3632, 35sylibrd 259 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) → ((1st𝑦) ∪ (2nd𝑦)) ∈ (𝐶m (𝐴𝐵))))
37 1st2nd2 8052 . . . . . . 7 (𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
3837ad2antll 729 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
3925adantrl 716 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (1st𝑦):𝐴𝐶)
4029adantrl 716 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (2nd𝑦):𝐵𝐶)
41 res0 6004 . . . . . . . . . 10 ((1st𝑦) ↾ ∅) = ∅
42 res0 6004 . . . . . . . . . 10 ((2nd𝑦) ↾ ∅) = ∅
4341, 42eqtr4i 2766 . . . . . . . . 9 ((1st𝑦) ↾ ∅) = ((2nd𝑦) ↾ ∅)
44 simplr 769 . . . . . . . . . 10 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (𝐴𝐵) = ∅)
4544reseq2d 6000 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → ((1st𝑦) ↾ (𝐴𝐵)) = ((1st𝑦) ↾ ∅))
4644reseq2d 6000 . . . . . . . . 9 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → ((2nd𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ ∅))
4743, 45, 463eqtr4a 2801 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵)))
48 fresaunres1 6782 . . . . . . . 8 (((1st𝑦):𝐴𝐶 ∧ (2nd𝑦):𝐵𝐶 ∧ ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴) = (1st𝑦))
4939, 40, 47, 48syl3anc 1370 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴) = (1st𝑦))
50 fresaunres2 6781 . . . . . . . 8 (((1st𝑦):𝐴𝐶 ∧ (2nd𝑦):𝐵𝐶 ∧ ((1st𝑦) ↾ (𝐴𝐵)) = ((2nd𝑦) ↾ (𝐴𝐵))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵) = (2nd𝑦))
5139, 40, 47, 50syl3anc 1370 . . . . . . 7 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵) = (2nd𝑦))
5249, 51opeq12d 4886 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩ = ⟨(1st𝑦), (2nd𝑦)⟩)
5338, 52eqtr4d 2778 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → 𝑦 = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩)
54 reseq1 5994 . . . . . . 7 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑥𝐴) = (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴))
55 reseq1 5994 . . . . . . 7 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑥𝐵) = (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵))
5654, 55opeq12d 4886 . . . . . 6 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → ⟨(𝑥𝐴), (𝑥𝐵)⟩ = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩)
5756eqeq2d 2746 . . . . 5 (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ ↔ 𝑦 = ⟨(((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐴), (((1st𝑦) ∪ (2nd𝑦)) ↾ 𝐵)⟩))
5853, 57syl5ibrcom 247 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) → 𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩))
59 ffn 6737 . . . . . . . 8 (𝑥:(𝐴𝐵)⟶𝐶𝑥 Fn (𝐴𝐵))
60 fnresdm 6688 . . . . . . . 8 (𝑥 Fn (𝐴𝐵) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
615, 59, 603syl 18 . . . . . . 7 (𝑥 ∈ (𝐶m (𝐴𝐵)) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
6261ad2antrl 728 . . . . . 6 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (𝑥 ↾ (𝐴𝐵)) = 𝑥)
6362eqcomd 2741 . . . . 5 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → 𝑥 = (𝑥 ↾ (𝐴𝐵)))
64 vex 3482 . . . . . . . . . 10 𝑥 ∈ V
6564resex 6049 . . . . . . . . 9 (𝑥𝐴) ∈ V
6664resex 6049 . . . . . . . . 9 (𝑥𝐵) ∈ V
6765, 66op1std 8023 . . . . . . . 8 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (1st𝑦) = (𝑥𝐴))
6865, 66op2ndd 8024 . . . . . . . 8 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (2nd𝑦) = (𝑥𝐵))
6967, 68uneq12d 4179 . . . . . . 7 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → ((1st𝑦) ∪ (2nd𝑦)) = ((𝑥𝐴) ∪ (𝑥𝐵)))
70 resundi 6014 . . . . . . 7 (𝑥 ↾ (𝐴𝐵)) = ((𝑥𝐴) ∪ (𝑥𝐵))
7169, 70eqtr4di 2793 . . . . . 6 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → ((1st𝑦) ∪ (2nd𝑦)) = (𝑥 ↾ (𝐴𝐵)))
7271eqeq2d 2746 . . . . 5 (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) ↔ 𝑥 = (𝑥 ↾ (𝐴𝐵))))
7363, 72syl5ibrcom 247 . . . 4 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩ → 𝑥 = ((1st𝑦) ∪ (2nd𝑦))))
7458, 73impbid 212 . . 3 ((((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) ∧ (𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵)))) → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) ↔ 𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩))
7574ex 412 . 2 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → ((𝑥 ∈ (𝐶m (𝐴𝐵)) ∧ 𝑦 ∈ ((𝐶m 𝐴) × (𝐶m 𝐵))) → (𝑥 = ((1st𝑦) ∪ (2nd𝑦)) ↔ 𝑦 = ⟨(𝑥𝐴), (𝑥𝐵)⟩)))
761, 4, 21, 36, 75en3d 9028 1 (((𝐴𝑉𝐵𝑊𝐶𝑋) ∧ (𝐴𝐵) = ∅) → (𝐶m (𝐴𝐵)) ≈ ((𝐶m 𝐴) × (𝐶m 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  cun 3961  cin 3962  wss 3963  c0 4339  cop 4637   class class class wbr 5148   × cxp 5687  cres 5691   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  m cmap 8865  cen 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-en 8985
This theorem is referenced by:  map2xp  9186  mapdom2  9187  mapdjuen  10219  ackbij1lem5  10261  hashmap  14471  mpct  45144
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