Theorem List for Intuitionistic Logic Explorer - 6501-6600 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | pmex 6501* |
The class of all partial functions from one set to another is a set.
(Contributed by NM, 15-Nov-2007.)
|
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ (Fun 𝑓 ∧ 𝑓 ⊆ (𝐴 × 𝐵))} ∈ V) |
|
Theorem | mapex 6502* |
The class of all functions mapping one set to another is a set. Remark
after Definition 10.24 of [Kunen] p. 31.
(Contributed by Raph Levien,
4-Dec-2003.)
|
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V) |
|
Theorem | fnmap 6503 |
Set exponentiation has a universal domain. (Contributed by NM,
8-Dec-2003.) (Revised by Mario Carneiro, 8-Sep-2013.)
|
⊢ ↑𝑚 Fn (V ×
V) |
|
Theorem | fnpm 6504 |
Partial function exponentiation has a universal domain. (Contributed by
Mario Carneiro, 14-Nov-2013.)
|
⊢ ↑pm Fn (V ×
V) |
|
Theorem | reldmmap 6505 |
Set exponentiation is a well-behaved binary operator. (Contributed by
Stefan O'Rear, 27-Feb-2015.)
|
⊢ Rel dom
↑𝑚 |
|
Theorem | mapvalg 6506* |
The value of set exponentiation. (𝐴 ↑𝑚 𝐵) is the set of all
functions that map from 𝐵 to 𝐴. Definition 10.24 of
[Kunen]
p. 24. (Contributed by NM, 8-Dec-2003.) (Revised by Mario Carneiro,
8-Sep-2013.)
|
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ↑𝑚 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴}) |
|
Theorem | pmvalg 6507* |
The value of the partial mapping operation. (𝐴 ↑pm 𝐵) is the set
of all partial functions that map from 𝐵 to 𝐴.
(Contributed by
NM, 15-Nov-2007.) (Revised by Mario Carneiro, 8-Sep-2013.)
|
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ↑pm 𝐵) = {𝑓 ∈ 𝒫 (𝐵 × 𝐴) ∣ Fun 𝑓}) |
|
Theorem | mapval 6508* |
The value of set exponentiation (inference version). (𝐴 ↑𝑚
𝐵) is
the set of all functions that map from 𝐵 to 𝐴.
Definition
10.24 of [Kunen] p. 24. (Contributed by
NM, 8-Dec-2003.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈
V ⇒ ⊢ (𝐴 ↑𝑚 𝐵) = {𝑓 ∣ 𝑓:𝐵⟶𝐴} |
|
Theorem | elmapg 6509 |
Membership relation for set exponentiation. (Contributed by NM,
17-Oct-2006.) (Revised by Mario Carneiro, 15-Nov-2014.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐶:𝐵⟶𝐴)) |
|
Theorem | elmapd 6510 |
Deduction form of elmapg 6509. (Contributed by BJ, 11-Apr-2020.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐶:𝐵⟶𝐴)) |
|
Theorem | mapdm0 6511 |
The empty set is the only map with empty domain. (Contributed by Glauco
Siliprandi, 11-Oct-2020.) (Proof shortened by Thierry Arnoux,
3-Dec-2021.)
|
⊢ (𝐵 ∈ 𝑉 → (𝐵 ↑𝑚 ∅) =
{∅}) |
|
Theorem | elpmg 6512 |
The predicate "is a partial function." (Contributed by Mario
Carneiro,
14-Nov-2013.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝐶 ∧ 𝐶 ⊆ (𝐵 × 𝐴)))) |
|
Theorem | elpm2g 6513 |
The predicate "is a partial function." (Contributed by NM,
31-Dec-2013.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵))) |
|
Theorem | elpm2r 6514 |
Sufficient condition for being a partial function. (Contributed by NM,
31-Dec-2013.)
|
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐹:𝐶⟶𝐴 ∧ 𝐶 ⊆ 𝐵)) → 𝐹 ∈ (𝐴 ↑pm 𝐵)) |
|
Theorem | elpmi 6515 |
A partial function is a function. (Contributed by Mario Carneiro,
15-Sep-2015.)
|
⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵)) |
|
Theorem | pmfun 6516 |
A partial function is a function. (Contributed by Mario Carneiro,
30-Jan-2014.) (Revised by Mario Carneiro, 26-Apr-2015.)
|
⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) → Fun 𝐹) |
|
Theorem | elmapex 6517 |
Eliminate antecedent for mapping theorems: domain can be taken to be a
set. (Contributed by Stefan O'Rear, 8-Oct-2014.)
|
⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → (𝐵 ∈ V ∧ 𝐶 ∈ V)) |
|
Theorem | elmapi 6518 |
A mapping is a function, forward direction only with superfluous
antecedent removed. (Contributed by Stefan O'Rear, 10-Oct-2014.)
|
⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → 𝐴:𝐶⟶𝐵) |
|
Theorem | elmapfn 6519 |
A mapping is a function with the appropriate domain. (Contributed by AV,
6-Apr-2019.)
|
⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → 𝐴 Fn 𝐶) |
|
Theorem | elmapfun 6520 |
A mapping is always a function. (Contributed by Stefan O'Rear,
9-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
|
⊢ (𝐴 ∈ (𝐵 ↑𝑚 𝐶) → Fun 𝐴) |
|
Theorem | elmapssres 6521 |
A restricted mapping is a mapping. (Contributed by Stefan O'Rear,
9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)
|
⊢ ((𝐴 ∈ (𝐵 ↑𝑚 𝐶) ∧ 𝐷 ⊆ 𝐶) → (𝐴 ↾ 𝐷) ∈ (𝐵 ↑𝑚 𝐷)) |
|
Theorem | fpmg 6522 |
A total function is a partial function. (Contributed by Mario Carneiro,
31-Dec-2013.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → 𝐹 ∈ (𝐵 ↑pm 𝐴)) |
|
Theorem | pmss12g 6523 |
Subset relation for the set of partial functions. (Contributed by Mario
Carneiro, 31-Dec-2013.)
|
⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → (𝐴 ↑pm 𝐵) ⊆ (𝐶 ↑pm 𝐷)) |
|
Theorem | pmresg 6524 |
Elementhood of a restricted function in the set of partial functions.
(Contributed by Mario Carneiro, 31-Dec-2013.)
|
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐹 ∈ (𝐴 ↑pm 𝐶)) → (𝐹 ↾ 𝐵) ∈ (𝐴 ↑pm 𝐵)) |
|
Theorem | elmap 6525 |
Membership relation for set exponentiation. (Contributed by NM,
8-Dec-2003.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈
V ⇒ ⊢ (𝐹 ∈ (𝐴 ↑𝑚 𝐵) ↔ 𝐹:𝐵⟶𝐴) |
|
Theorem | mapval2 6526* |
Alternate expression for the value of set exponentiation. (Contributed
by NM, 3-Nov-2007.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈
V ⇒ ⊢ (𝐴 ↑𝑚 𝐵) = (𝒫 (𝐵 × 𝐴) ∩ {𝑓 ∣ 𝑓 Fn 𝐵}) |
|
Theorem | elpm 6527 |
The predicate "is a partial function." (Contributed by NM,
15-Nov-2007.) (Revised by Mario Carneiro, 14-Nov-2013.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈
V ⇒ ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (Fun 𝐹 ∧ 𝐹 ⊆ (𝐵 × 𝐴))) |
|
Theorem | elpm2 6528 |
The predicate "is a partial function." (Contributed by NM,
15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈
V ⇒ ⊢ (𝐹 ∈ (𝐴 ↑pm 𝐵) ↔ (𝐹:dom 𝐹⟶𝐴 ∧ dom 𝐹 ⊆ 𝐵)) |
|
Theorem | fpm 6529 |
A total function is a partial function. (Contributed by NM,
15-Nov-2007.) (Revised by Mario Carneiro, 31-Dec-2013.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈
V ⇒ ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ∈ (𝐵 ↑pm 𝐴)) |
|
Theorem | mapsspm 6530 |
Set exponentiation is a subset of partial maps. (Contributed by NM,
15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.)
|
⊢ (𝐴 ↑𝑚 𝐵) ⊆ (𝐴 ↑pm 𝐵) |
|
Theorem | pmsspw 6531 |
Partial maps are a subset of the power set of the Cartesian product of
its arguments. (Contributed by Mario Carneiro, 2-Jan-2017.)
|
⊢ (𝐴 ↑pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) |
|
Theorem | mapsspw 6532 |
Set exponentiation is a subset of the power set of the Cartesian product
of its arguments. (Contributed by NM, 8-Dec-2006.) (Revised by Mario
Carneiro, 26-Apr-2015.)
|
⊢ (𝐴 ↑𝑚 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) |
|
Theorem | fvmptmap 6533* |
Special case of fvmpt 5452 for operator theorems. (Contributed by NM,
27-Nov-2007.)
|
⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V & ⊢ 𝑅 ∈ V & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)
& ⊢ 𝐹 = (𝑥 ∈ (𝑅 ↑𝑚 𝐷) ↦ 𝐵) ⇒ ⊢ (𝐴:𝐷⟶𝑅 → (𝐹‘𝐴) = 𝐶) |
|
Theorem | map0e 6534 |
Set exponentiation with an empty exponent (ordinal number 0) is ordinal
number 1. Exercise 4.42(a) of [Mendelson] p. 255. (Contributed by NM,
10-Dec-2003.) (Revised by Mario Carneiro, 30-Apr-2015.)
|
⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑𝑚 ∅) =
1o) |
|
Theorem | map0b 6535 |
Set exponentiation with an empty base is the empty set, provided the
exponent is nonempty. Theorem 96 of [Suppes] p. 89. (Contributed by
NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
|
⊢ (𝐴 ≠ ∅ → (∅
↑𝑚 𝐴) = ∅) |
|
Theorem | map0g 6536 |
Set exponentiation is empty iff the base is empty and the exponent is
not empty. Theorem 97 of [Suppes] p. 89.
(Contributed by Mario
Carneiro, 30-Apr-2015.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ↑𝑚 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠
∅))) |
|
Theorem | map0 6537 |
Set exponentiation is empty iff the base is empty and the exponent is
not empty. Theorem 97 of [Suppes] p. 89.
(Contributed by NM,
10-Dec-2003.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈
V ⇒ ⊢ ((𝐴 ↑𝑚 𝐵) = ∅ ↔ (𝐴 = ∅ ∧ 𝐵 ≠
∅)) |
|
Theorem | mapsn 6538* |
The value of set exponentiation with a singleton exponent. Theorem 98
of [Suppes] p. 89. (Contributed by NM,
10-Dec-2003.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈
V ⇒ ⊢ (𝐴 ↑𝑚 {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {〈𝐵, 𝑦〉}} |
|
Theorem | mapss 6539 |
Subset inheritance for set exponentiation. Theorem 99 of [Suppes]
p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro,
26-Apr-2015.)
|
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐴 ⊆ 𝐵) → (𝐴 ↑𝑚 𝐶) ⊆ (𝐵 ↑𝑚 𝐶)) |
|
Theorem | fdiagfn 6540* |
Functionality of the diagonal map. (Contributed by Stefan O'Rear,
24-Jan-2015.)
|
⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) ⇒ ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐵⟶(𝐵 ↑𝑚 𝐼)) |
|
Theorem | fvdiagfn 6541* |
Functionality of the diagonal map. (Contributed by Stefan O'Rear,
24-Jan-2015.)
|
⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥})) ⇒ ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (𝐼 × {𝑋})) |
|
Theorem | mapsnconst 6542 |
Every singleton map is a constant function. (Contributed by Stefan
O'Rear, 25-Mar-2015.)
|
⊢ 𝑆 = {𝑋}
& ⊢ 𝐵 ∈ V & ⊢ 𝑋 ∈
V ⇒ ⊢ (𝐹 ∈ (𝐵 ↑𝑚 𝑆) → 𝐹 = (𝑆 × {(𝐹‘𝑋)})) |
|
Theorem | mapsncnv 6543* |
Expression for the inverse of the canonical map between a set and its
set of singleton functions. (Contributed by Stefan O'Rear,
21-Mar-2015.)
|
⊢ 𝑆 = {𝑋}
& ⊢ 𝐵 ∈ V & ⊢ 𝑋 ∈ V & ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)) ⇒ ⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) |
|
Theorem | mapsnf1o2 6544* |
Explicit bijection between a set and its singleton functions.
(Contributed by Stefan O'Rear, 21-Mar-2015.)
|
⊢ 𝑆 = {𝑋}
& ⊢ 𝐵 ∈ V & ⊢ 𝑋 ∈ V & ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)) ⇒ ⊢ 𝐹:(𝐵 ↑𝑚 𝑆)–1-1-onto→𝐵 |
|
Theorem | mapsnf1o3 6545* |
Explicit bijection in the reverse of mapsnf1o2 6544. (Contributed by
Stefan O'Rear, 24-Mar-2015.)
|
⊢ 𝑆 = {𝑋}
& ⊢ 𝐵 ∈ V & ⊢ 𝑋 ∈ V & ⊢ 𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) ⇒ ⊢ 𝐹:𝐵–1-1-onto→(𝐵 ↑𝑚 𝑆) |
|
2.6.26 Infinite Cartesian products
|
|
Syntax | cixp 6546 |
Extend class notation to include infinite Cartesian products.
|
class X𝑥 ∈ 𝐴 𝐵 |
|
Definition | df-ixp 6547* |
Definition of infinite Cartesian product of [Enderton] p. 54. Enderton
uses a bold "X" with 𝑥 ∈ 𝐴 written underneath or as a
subscript, as
does Stoll p. 47. Some books use a capital pi, but we will reserve that
notation for products of numbers. Usually 𝐵 represents a class
expression containing 𝑥 free and thus can be thought of as
𝐵(𝑥). Normally, 𝑥 is not free in 𝐴,
although this is
not a requirement of the definition. (Contributed by NM,
28-Sep-2006.)
|
⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} |
|
Theorem | dfixp 6548* |
Eliminate the expression {𝑥 ∣ 𝑥 ∈ 𝐴} in df-ixp 6547, under the
assumption that 𝐴 and 𝑥 are disjoint. This way,
we can say that
𝑥 is bound in X𝑥 ∈
𝐴𝐵 even if it appears free in 𝐴.
(Contributed by Mario Carneiro, 12-Aug-2016.)
|
⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} |
|
Theorem | ixpsnval 6549* |
The value of an infinite Cartesian product with a singleton.
(Contributed by AV, 3-Dec-2018.)
|
⊢ (𝑋 ∈ 𝑉 → X𝑥 ∈ {𝑋}𝐵 = {𝑓 ∣ (𝑓 Fn {𝑋} ∧ (𝑓‘𝑋) ∈ ⦋𝑋 / 𝑥⦌𝐵)}) |
|
Theorem | elixp2 6550* |
Membership in an infinite Cartesian product. See df-ixp 6547 for
discussion of the notation. (Contributed by NM, 28-Sep-2006.)
|
⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
|
Theorem | fvixp 6551* |
Projection of a factor of an indexed Cartesian product. (Contributed by
Mario Carneiro, 11-Jun-2016.)
|
⊢ (𝑥 = 𝐶 → 𝐵 = 𝐷) ⇒ ⊢ ((𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐷) |
|
Theorem | ixpfn 6552* |
A nuple is a function. (Contributed by FL, 6-Jun-2011.) (Revised by
Mario Carneiro, 31-May-2014.)
|
⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴) |
|
Theorem | elixp 6553* |
Membership in an infinite Cartesian product. (Contributed by NM,
28-Sep-2006.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
|
Theorem | elixpconst 6554* |
Membership in an infinite Cartesian product of a constant 𝐵.
(Contributed by NM, 12-Apr-2008.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ 𝐹:𝐴⟶𝐵) |
|
Theorem | ixpconstg 6555* |
Infinite Cartesian product of a constant 𝐵. (Contributed by Mario
Carneiro, 11-Jan-2015.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → X𝑥 ∈ 𝐴 𝐵 = (𝐵 ↑𝑚 𝐴)) |
|
Theorem | ixpconst 6556* |
Infinite Cartesian product of a constant 𝐵. (Contributed by NM,
28-Sep-2006.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈
V ⇒ ⊢ X𝑥 ∈ 𝐴 𝐵 = (𝐵 ↑𝑚 𝐴) |
|
Theorem | ixpeq1 6557* |
Equality theorem for infinite Cartesian product. (Contributed by NM,
29-Sep-2006.)
|
⊢ (𝐴 = 𝐵 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶) |
|
Theorem | ixpeq1d 6558* |
Equality theorem for infinite Cartesian product. (Contributed by Mario
Carneiro, 11-Jun-2016.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶) |
|
Theorem | ss2ixp 6559 |
Subclass theorem for infinite Cartesian product. (Contributed by NM,
29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.)
|
⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶) |
|
Theorem | ixpeq2 6560 |
Equality theorem for infinite Cartesian product. (Contributed by NM,
29-Sep-2006.)
|
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) |
|
Theorem | ixpeq2dva 6561* |
Equality theorem for infinite Cartesian product. (Contributed by Mario
Carneiro, 11-Jun-2016.)
|
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) |
|
Theorem | ixpeq2dv 6562* |
Equality theorem for infinite Cartesian product. (Contributed by Mario
Carneiro, 11-Jun-2016.)
|
⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) |
|
Theorem | cbvixp 6563* |
Change bound variable in an indexed Cartesian product. (Contributed by
Jeff Madsen, 20-Jun-2011.)
|
⊢ Ⅎ𝑦𝐵
& ⊢ Ⅎ𝑥𝐶
& ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶 |
|
Theorem | cbvixpv 6564* |
Change bound variable in an indexed Cartesian product. (Contributed by
Jeff Madsen, 2-Sep-2009.)
|
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ X𝑥 ∈ 𝐴 𝐵 = X𝑦 ∈ 𝐴 𝐶 |
|
Theorem | nfixpxy 6565* |
Bound-variable hypothesis builder for indexed Cartesian product.
(Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Jim Kingdon,
15-Feb-2023.)
|
⊢ Ⅎ𝑦𝐴
& ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 |
|
Theorem | nfixp1 6566 |
The index variable in an indexed Cartesian product is not free.
(Contributed by Jeff Madsen, 19-Jun-2011.) (Revised by Mario Carneiro,
15-Oct-2016.)
|
⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵 |
|
Theorem | ixpprc 6567* |
A cartesian product of proper-class many sets is empty, because any
function in the cartesian product has to be a set with domain 𝐴,
which is not possible for a proper class domain. (Contributed by Mario
Carneiro, 25-Jan-2015.)
|
⊢ (¬ 𝐴 ∈ V → X𝑥 ∈
𝐴 𝐵 = ∅) |
|
Theorem | ixpf 6568* |
A member of an infinite Cartesian product maps to the indexed union of
the product argument. Remark in [Enderton] p. 54. (Contributed by NM,
28-Sep-2006.)
|
⊢ (𝐹 ∈ X𝑥 ∈ 𝐴 𝐵 → 𝐹:𝐴⟶∪
𝑥 ∈ 𝐴 𝐵) |
|
Theorem | uniixp 6569* |
The union of an infinite Cartesian product is included in a Cartesian
product. (Contributed by NM, 28-Sep-2006.) (Revised by Mario Carneiro,
24-Jun-2015.)
|
⊢ ∪ X𝑥 ∈
𝐴 𝐵 ⊆ (𝐴 × ∪ 𝑥 ∈ 𝐴 𝐵) |
|
Theorem | ixpexgg 6570* |
The existence of an infinite Cartesian product. 𝑥 is normally a
free-variable parameter in 𝐵. Remark in Enderton p. 54.
(Contributed by NM, 28-Sep-2006.) (Revised by Jim Kingdon,
15-Feb-2023.)
|
⊢ ((𝐴 ∈ 𝑊 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) → X𝑥 ∈ 𝐴 𝐵 ∈ V) |
|
Theorem | ixpin 6571* |
The intersection of two infinite Cartesian products. (Contributed by
Mario Carneiro, 3-Feb-2015.)
|
⊢ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶) |
|
Theorem | ixpiinm 6572* |
The indexed intersection of a collection of infinite Cartesian products.
(Contributed by Mario Carneiro, 6-Feb-2015.) (Revised by Jim Kingdon,
15-Feb-2023.)
|
⊢ (∃𝑧 𝑧 ∈ 𝐵 → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 = ∩
𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶) |
|
Theorem | ixpintm 6573* |
The intersection of a collection of infinite Cartesian products.
(Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Jim Kingdon,
15-Feb-2023.)
|
⊢ (∃𝑧 𝑧 ∈ 𝐵 → X𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦) |
|
Theorem | ixp0x 6574 |
An infinite Cartesian product with an empty index set. (Contributed by
NM, 21-Sep-2007.)
|
⊢ X𝑥 ∈ ∅ 𝐴 = {∅} |
|
Theorem | ixpssmap2g 6575* |
An infinite Cartesian product is a subset of set exponentiation. This
version of ixpssmapg 6576 avoids ax-coll 4003. (Contributed by Mario
Carneiro, 16-Nov-2014.)
|
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴)) |
|
Theorem | ixpssmapg 6576* |
An infinite Cartesian product is a subset of set exponentiation.
(Contributed by Jeff Madsen, 19-Jun-2011.)
|
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴)) |
|
Theorem | 0elixp 6577 |
Membership of the empty set in an infinite Cartesian product.
(Contributed by Steve Rodriguez, 29-Sep-2006.)
|
⊢ ∅ ∈ X𝑥 ∈ ∅ 𝐴 |
|
Theorem | ixpm 6578* |
If an infinite Cartesian product of a family 𝐵(𝑥) is inhabited,
every 𝐵(𝑥) is inhabited. (Contributed by Mario
Carneiro,
22-Jun-2016.) (Revised by Jim Kingdon, 16-Feb-2023.)
|
⊢ (∃𝑓 𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 → ∀𝑥 ∈ 𝐴 ∃𝑧 𝑧 ∈ 𝐵) |
|
Theorem | ixp0 6579 |
The infinite Cartesian product of a family 𝐵(𝑥) with an empty
member is empty. (Contributed by NM, 1-Oct-2006.) (Revised by Jim
Kingdon, 16-Feb-2023.)
|
⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈
𝐴 𝐵 = ∅) |
|
Theorem | ixpssmap 6580* |
An infinite Cartesian product is a subset of set exponentiation. Remark
in [Enderton] p. 54. (Contributed by
NM, 28-Sep-2006.)
|
⊢ 𝐵 ∈ V ⇒ ⊢ X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑𝑚 𝐴) |
|
Theorem | resixp 6581* |
Restriction of an element of an infinite Cartesian product.
(Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro,
31-May-2014.)
|
⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐹 ∈ X𝑥 ∈ 𝐴 𝐶) → (𝐹 ↾ 𝐵) ∈ X𝑥 ∈ 𝐵 𝐶) |
|
Theorem | mptelixpg 6582* |
Condition for an explicit member of an indexed product. (Contributed by
Stefan O'Rear, 4-Jan-2015.)
|
⊢ (𝐼 ∈ 𝑉 → ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾)) |
|
Theorem | elixpsn 6583* |
Membership in a class of singleton functions. (Contributed by Stefan
O'Rear, 24-Jan-2015.)
|
⊢ (𝐴 ∈ 𝑉 → (𝐹 ∈ X𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑦 ∈ 𝐵 𝐹 = {〈𝐴, 𝑦〉})) |
|
Theorem | ixpsnf1o 6584* |
A bijection between a class and single-point functions to it.
(Contributed by Stefan O'Rear, 24-Jan-2015.)
|
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ({𝐼} × {𝑥})) ⇒ ⊢ (𝐼 ∈ 𝑉 → 𝐹:𝐴–1-1-onto→X𝑦 ∈ {𝐼}𝐴) |
|
Theorem | mapsnf1o 6585* |
A bijection between a set and single-point functions to it.
(Contributed by Stefan O'Rear, 24-Jan-2015.)
|
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ ({𝐼} × {𝑥})) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹:𝐴–1-1-onto→(𝐴 ↑𝑚 {𝐼})) |
|
2.6.27 Equinumerosity
|
|
Syntax | cen 6586 |
Extend class definition to include the equinumerosity relation
("approximately equals" symbol)
|
class ≈ |
|
Syntax | cdom 6587 |
Extend class definition to include the dominance relation (curly
less-than-or-equal)
|
class ≼ |
|
Syntax | cfn 6588 |
Extend class definition to include the class of all finite sets.
|
class Fin |
|
Definition | df-en 6589* |
Define the equinumerosity relation. Definition of [Enderton] p. 129.
We define ≈ to be a binary relation rather
than a connective, so
its arguments must be sets to be meaningful. This is acceptable because
we do not consider equinumerosity for proper classes. We derive the
usual definition as bren 6595. (Contributed by NM, 28-Mar-1998.)
|
⊢ ≈ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1-onto→𝑦} |
|
Definition | df-dom 6590* |
Define the dominance relation. Compare Definition of [Enderton] p. 145.
Typical textbook definitions are derived as brdom 6598 and domen 6599.
(Contributed by NM, 28-Mar-1998.)
|
⊢ ≼ = {〈𝑥, 𝑦〉 ∣ ∃𝑓 𝑓:𝑥–1-1→𝑦} |
|
Definition | df-fin 6591* |
Define the (proper) class of all finite sets. Similar to Definition
10.29 of [TakeutiZaring] p. 91,
whose "Fin(a)" corresponds to
our "𝑎 ∈ Fin". This definition is
meaningful whether or not we
accept the Axiom of Infinity ax-inf2 12864. (Contributed by NM,
22-Aug-2008.)
|
⊢ Fin = {𝑥 ∣ ∃𝑦 ∈ ω 𝑥 ≈ 𝑦} |
|
Theorem | relen 6592 |
Equinumerosity is a relation. (Contributed by NM, 28-Mar-1998.)
|
⊢ Rel ≈ |
|
Theorem | reldom 6593 |
Dominance is a relation. (Contributed by NM, 28-Mar-1998.)
|
⊢ Rel ≼ |
|
Theorem | encv 6594 |
If two classes are equinumerous, both classes are sets. (Contributed by
AV, 21-Mar-2019.)
|
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) |
|
Theorem | bren 6595* |
Equinumerosity relation. (Contributed by NM, 15-Jun-1998.)
|
⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
|
Theorem | brdomg 6596* |
Dominance relation. (Contributed by NM, 15-Jun-1998.)
|
⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵)) |
|
Theorem | brdomi 6597* |
Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
|
⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) |
|
Theorem | brdom 6598* |
Dominance relation. (Contributed by NM, 15-Jun-1998.)
|
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1→𝐵) |
|
Theorem | domen 6599* |
Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146.
(Contributed by NM, 15-Jun-1998.)
|
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵)) |
|
Theorem | domeng 6600* |
Dominance in terms of equinumerosity, with the sethood requirement
expressed as an antecedent. Example 1 of [Enderton] p. 146.
(Contributed by NM, 24-Apr-2004.)
|
⊢ (𝐵 ∈ 𝐶 → (𝐴 ≼ 𝐵 ↔ ∃𝑥(𝐴 ≈ 𝑥 ∧ 𝑥 ⊆ 𝐵))) |