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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | cpm 8401 | Extend the definition of a class to include the partial mapping operation. (Read for 𝐴 ↑pm 𝐵, "the set of all partial functions that map from 𝐵 to 𝐴.) |
class ↑pm | ||
Definition | df-map 8402* | Define the mapping operation or set exponentiation. The set of all functions that map from 𝐵 to 𝐴 is written (𝐴 ↑m 𝐵) (see mapval 8412). Many authors write 𝐴 followed by 𝐵 as a superscript for this operation and rely on context to avoid confusion other exponentiation operations (e.g., Definition 10.42 of [TakeutiZaring] p. 95). Other authors show 𝐵 as a prefixed superscript, which is read "𝐴 pre 𝐵 " (e.g., definition of [Enderton] p. 52). Definition 8.21 of [Eisenberg] p. 125 uses the notation Map(𝐵, 𝐴) for our (𝐴 ↑m 𝐵). The up-arrow is used by Donald Knuth for iterated exponentiation (Science 194, 1235-1242, 1976). We adopt the first case of his notation (simple exponentiation) and subscript it with m to distinguish it from other kinds of exponentiation. (Contributed by NM, 8-Dec-2003.) |
⊢ ↑m = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∣ 𝑓:𝑦⟶𝑥}) | ||
Definition | df-pm 8403* | Define the partial mapping operation. A partial function from 𝐵 to 𝐴 is a function from a subset of 𝐵 to 𝐴. The set of all partial functions from 𝐵 to 𝐴 is written (𝐴 ↑pm 𝐵) (see pmvalg 8411). A notation for this operation apparently does not appear in the literature. We use ↑pm to distinguish it from the less general set exponentiation operation ↑m (df-map 8402). See mapsspm 8434 for its relationship to set exponentiation. (Contributed by NM, 15-Nov-2007.) |
⊢ ↑pm = (𝑥 ∈ V, 𝑦 ∈ V ↦ {𝑓 ∈ 𝒫 (𝑦 × 𝑥) ∣ Fun 𝑓}) | ||
Theorem | mapprc 8404* | When 𝐴 is a proper class, the class of all functions mapping 𝐴 to 𝐵 is empty. Exercise 4.41 of [Mendelson] p. 255. (Contributed by NM, 8-Dec-2003.) |
⊢ (¬ 𝐴 ∈ V → {𝑓 ∣ 𝑓:𝐴⟶𝐵} = ∅) | ||
Theorem | pmex 8405* | 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 8406* | 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 8407 | 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 8408 | Partial function exponentiation has a universal domain. (Contributed by Mario Carneiro, 14-Nov-2013.) |
⊢ ↑pm Fn (V × V) | ||
Theorem | reldmmap 8409 | Set exponentiation is a well-behaved binary operator. (Contributed by Stefan O'Rear, 27-Feb-2015.) |
⊢ Rel dom ↑m | ||
Theorem | mapvalg 8410* | 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 8411* | 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 8412* | 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 8413 | 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 8414 | Deduction form of elmapg 8413. (Contributed by BJ, 11-Apr-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐶 ∈ (𝐴 ↑m 𝐵) ↔ 𝐶:𝐵⟶𝐴)) | ||
Theorem | mapdm0 8415 | 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 8416 | 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 8417 | 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 8418 | 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 8419 | 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 8420 | 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 8421 | 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 8422 | 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 8423 | A mapping is a function with the appropriate domain. (Contributed by AV, 6-Apr-2019.) |
⊢ (𝐴 ∈ (𝐵 ↑m 𝐶) → 𝐴 Fn 𝐶) | ||
Theorem | elmapfun 8424 | 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 8425 | 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 8426 | 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 8427 | 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 8428 | 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 8429 | Membership relation for set exponentiation. (Contributed by NM, 8-Dec-2003.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐹 ∈ (𝐴 ↑m 𝐵) ↔ 𝐹:𝐵⟶𝐴) | ||
Theorem | mapval2 8430* | 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 8431 | 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 8432 | 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 8433 | 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 8434 | 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 8435 | 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 8436 | 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 | mapfvd 8437 | The value of a function that maps from 𝐵 to 𝐴. (Contributed by AV, 2-Feb-2023.) |
⊢ 𝑀 = (𝐴 ↑m 𝐵) & ⊢ (𝜑 → 𝐹 ∈ 𝑀) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹‘𝑋) ∈ 𝐴) | ||
Theorem | elmapresaun 8438 | fresaun 6544 transposed to mappings. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Stefan O'Rear, 6-May-2015.) |
⊢ ((𝐹 ∈ (𝐶 ↑m 𝐴) ∧ 𝐺 ∈ (𝐶 ↑m 𝐵) ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺) ∈ (𝐶 ↑m (𝐴 ∪ 𝐵))) | ||
Theorem | fvmptmap 8439* | Special case of fvmpt 6763 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 8440 | 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.) (Proof shortened by AV, 14-Jul-2022.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m ∅) = 1o) | ||
Theorem | map0b 8441 | 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 8442 | 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 | 0map0sn0 8443 | The set of mappings of the empty set to the empty set is the singleton containing the empty set. (Contributed by AV, 31-Mar-2024.) |
⊢ (∅ ↑m ∅) = {∅} | ||
Theorem | mapsnd 8444* | The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Glauco Siliprandi, 24-Dec-2020.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (𝐴 ↑m {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {〈𝐵, 𝑦〉}}) | ||
Theorem | map0 8445 | 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 8446* | The value of set exponentiation with a singleton exponent. Theorem 98 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Proof shortened by AV, 17-Jul-2022.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ↑m {𝐵}) = {𝑓 ∣ ∃𝑦 ∈ 𝐴 𝑓 = {〈𝐵, 𝑦〉}} | ||
Theorem | mapss 8447 | 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 8448* | 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 8449* | 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 8450 | 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 8451* | 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 8452* | 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 8453* | Explicit bijection in the reverse of mapsnf1o2 8452. (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) | ||
Theorem | ralxpmap 8454* | Quantification over functions in terms of quantification over values and punctured functions. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Stefan O'Rear, 5-May-2015.) |
⊢ (𝑓 = (𝑔 ∪ {〈𝐽, 𝑦〉}) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐽 ∈ 𝑇 → (∀𝑓 ∈ (𝑆 ↑m 𝑇)𝜑 ↔ ∀𝑦 ∈ 𝑆 ∀𝑔 ∈ (𝑆 ↑m (𝑇 ∖ {𝐽}))𝜓)) | ||
Syntax | cixp 8455 | Extend class notation to include infinite Cartesian products. |
class X𝑥 ∈ 𝐴 𝐵 | ||
Definition | df-ixp 8456* | 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 8457* | Eliminate the expression {𝑥 ∣ 𝑥 ∈ 𝐴} in df-ixp 8456, 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 8458* | 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 8459* | Membership in an infinite Cartesian product. See df-ixp 8456 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 8460* | 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 8461* | 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 8462* | 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 8463* | 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 8464* | 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 8465* | 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 8466* | Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) |
⊢ (𝐴 = 𝐵 → X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐵 𝐶) | ||
Theorem | ixpeq1d 8467* | 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 8468 | 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 8469 | Equality theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐶 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | ixpeq2dva 8470* | 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 8471* | 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 8472* | 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 8473* | 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 | nfixpw 8474* | Version of nfixp 8475 with a disjoint variable condition, which does not require ax-13 2386. (Contributed by Gino Giotto, 26-Jan-2024.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfixp 8475 | Bound-variable hypothesis builder for indexed Cartesian product. Usage of this theorem is discouraged because it depends on ax-13 2386. Use the weaker nfixpw 8474 when possible. (Contributed by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.) |
⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 ⇒ ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 | ||
Theorem | nfixp1 8476 | 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 8477* | 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 8478* | 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 8479* | 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 | ixpexg 8480* | 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 Mario Carneiro, 25-Jan-2015.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ∈ V) | ||
Theorem | ixpin 8481* | 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 | ixpiin 8482* | The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.) |
⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | ixpint 8483* | The intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.) |
⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝐵 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝑦) | ||
Theorem | ixp0x 8484 | An infinite Cartesian product with an empty index set. (Contributed by NM, 21-Sep-2007.) |
⊢ X𝑥 ∈ ∅ 𝐴 = {∅} | ||
Theorem | ixpssmap2g 8485* | An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 8486 avoids ax-rep 5183. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ (∪ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → X𝑥 ∈ 𝐴 𝐵 ⊆ (∪ 𝑥 ∈ 𝐴 𝐵 ↑m 𝐴)) | ||
Theorem | ixpssmapg 8486* | 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 8487 | Membership of the empty set in an infinite Cartesian product. (Contributed by Steve Rodriguez, 29-Sep-2006.) |
⊢ ∅ ∈ X𝑥 ∈ ∅ 𝐴 | ||
Theorem | ixpn0 8488 | The infinite Cartesian product of a family 𝐵(𝑥) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 9899. (Contributed by Mario Carneiro, 22-Jun-2016.) |
⊢ (X𝑥 ∈ 𝐴 𝐵 ≠ ∅ → ∀𝑥 ∈ 𝐴 𝐵 ≠ ∅) | ||
Theorem | ixp0 8489 | The infinite Cartesian product of a family 𝐵(𝑥) with an empty member is empty. The converse of this theorem is equivalent to the Axiom of Choice, see ac9 9899. (Contributed by NM, 1-Oct-2006.) (Proof shortened by Mario Carneiro, 22-Jun-2016.) |
⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → X𝑥 ∈ 𝐴 𝐵 = ∅) | ||
Theorem | ixpssmap 8490* | 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 8491* | 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 | undifixp 8492* | Union of two projections of a cartesian product. (Contributed by FL, 7-Nov-2011.) |
⊢ ((𝐹 ∈ X𝑥 ∈ 𝐵 𝐶 ∧ 𝐺 ∈ X𝑥 ∈ (𝐴 ∖ 𝐵)𝐶 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ∪ 𝐺) ∈ X𝑥 ∈ 𝐴 𝐶) | ||
Theorem | mptelixpg 8493* | 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 | resixpfo 8494* | Restriction of elements of an infinite Cartesian product creates a surjection, if the original Cartesian product is nonempty. (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ 𝐹 = (𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↦ (𝑓 ↾ 𝐵)) ⇒ ⊢ ((𝐵 ⊆ 𝐴 ∧ X𝑥 ∈ 𝐴 𝐶 ≠ ∅) → 𝐹:X𝑥 ∈ 𝐴 𝐶–onto→X𝑥 ∈ 𝐵 𝐶) | ||
Theorem | elixpsn 8495* | 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 8496* | 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 8497* | 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})) | ||
Theorem | boxriin 8498* | A rectangular subset of a rectangular set can be recovered as the relative intersection of single-axis restrictions. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ (∀𝑥 ∈ 𝐼 𝐴 ⊆ 𝐵 → X𝑥 ∈ 𝐼 𝐴 = (X𝑥 ∈ 𝐼 𝐵 ∩ ∩ 𝑦 ∈ 𝐼 X𝑥 ∈ 𝐼 if(𝑥 = 𝑦, 𝐴, 𝐵))) | ||
Theorem | boxcutc 8499* | The relative complement of a box set restricted on one axis. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝐶 ⊆ 𝐵) → (X𝑘 ∈ 𝐴 𝐵 ∖ X𝑘 ∈ 𝐴 if(𝑘 = 𝑋, 𝐶, 𝐵)) = X𝑘 ∈ 𝐴 if(𝑘 = 𝑋, (𝐵 ∖ 𝐶), 𝐵)) | ||
Syntax | cen 8500 | Extend class definition to include the equinumerosity relation ("approximately equals" symbol) |
class ≈ |
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