Home | Intuitionistic Logic Explorer Theorem List (p. 59 of 106) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | op1st 5801 | Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (1^{st} ‘⟨𝐴, 𝐵⟩) = 𝐴 | ||
Theorem | op2nd 5802 | Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (2^{nd} ‘⟨𝐴, 𝐵⟩) = 𝐵 | ||
Theorem | op1std 5803 | Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐶 = ⟨𝐴, 𝐵⟩ → (1^{st} ‘𝐶) = 𝐴) | ||
Theorem | op2ndd 5804 | Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐶 = ⟨𝐴, 𝐵⟩ → (2^{nd} ‘𝐶) = 𝐵) | ||
Theorem | op1stg 5805 | Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1^{st} ‘⟨𝐴, 𝐵⟩) = 𝐴) | ||
Theorem | op2ndg 5806 | Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2^{nd} ‘⟨𝐴, 𝐵⟩) = 𝐵) | ||
Theorem | ot1stg 5807 | Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 5807, ot2ndg 5808, ot3rdgg 5809.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1^{st} ‘(1^{st} ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴) | ||
Theorem | ot2ndg 5808 | Extract the second member of an ordered triple. (See ot1stg 5807 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (2^{nd} ‘(1^{st} ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵) | ||
Theorem | ot3rdgg 5809 | Extract the third member of an ordered triple. (See ot1stg 5807 comment.) (Contributed by NM, 3-Apr-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (2^{nd} ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶) | ||
Theorem | 1stval2 5810 | Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.) |
⊢ (𝐴 ∈ (V × V) → (1^{st} ‘𝐴) = ∩ ∩ 𝐴) | ||
Theorem | 2ndval2 5811 | Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.) |
⊢ (𝐴 ∈ (V × V) → (2^{nd} ‘𝐴) = ∩ ∩ ∩ ^{◡}{𝐴}) | ||
Theorem | fo1st 5812 | The 1^{st} function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ 1^{st} :V–onto→V | ||
Theorem | fo2nd 5813 | The 2^{nd} function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ 2^{nd} :V–onto→V | ||
Theorem | f1stres 5814 | Mapping of a restriction of the 1^{st} (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (1^{st} ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 | ||
Theorem | f2ndres 5815 | Mapping of a restriction of the 2^{nd} (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (2^{nd} ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 | ||
Theorem | fo1stresm 5816* | Onto mapping of a restriction of the 1^{st} (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.) |
⊢ (∃𝑦 𝑦 ∈ 𝐵 → (1^{st} ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴) | ||
Theorem | fo2ndresm 5817* | Onto mapping of a restriction of the 2^{nd} (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.) |
⊢ (∃𝑥 𝑥 ∈ 𝐴 → (2^{nd} ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵) | ||
Theorem | 1stcof 5818 | Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.) |
⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1^{st} ∘ 𝐹):𝐴⟶𝐵) | ||
Theorem | 2ndcof 5819 | Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.) |
⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2^{nd} ∘ 𝐹):𝐴⟶𝐶) | ||
Theorem | xp1st 5820 | Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1^{st} ‘𝐴) ∈ 𝐵) | ||
Theorem | xp2nd 5821 | Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → (2^{nd} ‘𝐴) ∈ 𝐶) | ||
Theorem | 1stexg 5822 | Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
⊢ (𝐴 ∈ 𝑉 → (1^{st} ‘𝐴) ∈ V) | ||
Theorem | 2ndexg 5823 | Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
⊢ (𝐴 ∈ 𝑉 → (2^{nd} ‘𝐴) ∈ V) | ||
Theorem | elxp6 5824 | Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4836. (Contributed by NM, 9-Oct-2004.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1^{st} ‘𝐴), (2^{nd} ‘𝐴)⟩ ∧ ((1^{st} ‘𝐴) ∈ 𝐵 ∧ (2^{nd} ‘𝐴) ∈ 𝐶))) | ||
Theorem | elxp7 5825 | Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4836. (Contributed by NM, 19-Aug-2006.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1^{st} ‘𝐴) ∈ 𝐵 ∧ (2^{nd} ‘𝐴) ∈ 𝐶))) | ||
Theorem | eqopi 5826 | Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.) |
⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1^{st} ‘𝐴) = 𝐵 ∧ (2^{nd} ‘𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩) | ||
Theorem | xp2 5827* | Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.) |
⊢ (𝐴 × 𝐵) = {𝑥 ∈ (V × V) ∣ ((1^{st} ‘𝑥) ∈ 𝐴 ∧ (2^{nd} ‘𝑥) ∈ 𝐵)} | ||
Theorem | unielxp 5828 | The membership relation for a cross product is inherited by union. (Contributed by NM, 16-Sep-2006.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶)) | ||
Theorem | 1st2nd2 5829 | Reconstruction of a member of a cross product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = ⟨(1^{st} ‘𝐴), (2^{nd} ‘𝐴)⟩) | ||
Theorem | xpopth 5830 | An ordered pair theorem for members of cross products. (Contributed by NM, 20-Jun-2007.) |
⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1^{st} ‘𝐴) = (1^{st} ‘𝐵) ∧ (2^{nd} ‘𝐴) = (2^{nd} ‘𝐵)) ↔ 𝐴 = 𝐵)) | ||
Theorem | eqop 5831 | Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.) |
⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ((1^{st} ‘𝐴) = 𝐵 ∧ (2^{nd} ‘𝐴) = 𝐶))) | ||
Theorem | eqop2 5832 | Two ways to express equality with an ordered pair. (Contributed by NM, 25-Feb-2014.) |
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐴 = ⟨𝐵, 𝐶⟩ ↔ (𝐴 ∈ (V × V) ∧ ((1^{st} ‘𝐴) = 𝐵 ∧ (2^{nd} ‘𝐴) = 𝐶))) | ||
Theorem | op1steq 5833* | Two ways of expressing that an element is the first member of an ordered pair. (Contributed by NM, 22-Sep-2013.) (Revised by Mario Carneiro, 23-Feb-2014.) |
⊢ (𝐴 ∈ (𝑉 × 𝑊) → ((1^{st} ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = ⟨𝐵, 𝑥⟩)) | ||
Theorem | 2nd1st 5834 | Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ^{◡}{𝐴} = ⟨(2^{nd} ‘𝐴), (1^{st} ‘𝐴)⟩) | ||
Theorem | 1st2nd 5835 | Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.) |
⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = ⟨(1^{st} ‘𝐴), (2^{nd} ‘𝐴)⟩) | ||
Theorem | 1stdm 5836 | The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.) |
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1^{st} ‘𝐴) ∈ dom 𝑅) | ||
Theorem | 2ndrn 5837 | The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.) |
⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2^{nd} ‘𝐴) ∈ ran 𝑅) | ||
Theorem | 1st2ndbr 5838 | Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.) |
⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → (1^{st} ‘𝐴)𝐵(2^{nd} ‘𝐴)) | ||
Theorem | releldm2 5839* | Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
⊢ (Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1^{st} ‘𝑥) = 𝐵)) | ||
Theorem | reldm 5840* | An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1^{st} ‘𝑥))) | ||
Theorem | sbcopeq1a 5841 | Equality theorem for substitution of a class for an ordered pair (analog of sbceq1a 2796 that avoids the existential quantifiers of copsexg 4009). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ([(1^{st} ‘𝐴) / 𝑥][(2^{nd} ‘𝐴) / 𝑦]𝜑 ↔ 𝜑)) | ||
Theorem | csbopeq1a 5842 | Equality theorem for substitution of a class 𝐴 for an ordered pair ⟨𝑥, 𝑦⟩ in 𝐵 (analog of csbeq1a 2888). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ⦋(1^{st} ‘𝐴) / 𝑥⦌⦋(2^{nd} ‘𝐴) / 𝑦⦌𝐵 = 𝐵) | ||
Theorem | dfopab2 5843* | A way to define an ordered-pair class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑧 ∈ (V × V) ∣ [(1^{st} ‘𝑧) / 𝑥][(2^{nd} ‘𝑧) / 𝑦]𝜑} | ||
Theorem | dfoprab3s 5844* | A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1^{st} ‘𝑤) / 𝑥][(2^{nd} ‘𝑤) / 𝑦]𝜑)} | ||
Theorem | dfoprab3 5845* | Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.) |
⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓)) ⇒ ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} | ||
Theorem | dfoprab4 5846* | Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 3-Sep-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓)) ⇒ ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} | ||
Theorem | dfoprab4f 5847* | Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓)) ⇒ ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} | ||
Theorem | dfxp3 5848* | Define the cross product of three classes. Compare df-xp 4379. (Contributed by FL, 6-Nov-2013.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) |
⊢ ((𝐴 × 𝐵) × 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶)} | ||
Theorem | elopabi 5849* | A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.) |
⊢ (𝑥 = (1^{st} ‘𝐴) → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = (2^{nd} ‘𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒) | ||
Theorem | eloprabi 5850* | A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.) |
⊢ (𝑥 = (1^{st} ‘(1^{st} ‘𝐴)) → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = (2^{nd} ‘(1^{st} ‘𝐴)) → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = (2^{nd} ‘𝐴) → (𝜒 ↔ 𝜃)) ⇒ ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝜃) | ||
Theorem | mpt2mptsx 5851* | Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Dec-2016.) |
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ ⦋(1^{st} ‘𝑧) / 𝑥⦌⦋(2^{nd} ‘𝑧) / 𝑦⦌𝐶) | ||
Theorem | mpt2mpts 5852* | Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 24-Sep-2015.) |
⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ (𝐴 × 𝐵) ↦ ⦋(1^{st} ‘𝑧) / 𝑥⦌⦋(2^{nd} ‘𝑧) / 𝑦⦌𝐶) | ||
Theorem | dmmpt2ssx 5853* | The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) | ||
Theorem | fmpt2x 5854* | Functionality, domain and codomain of a class given by the "maps to" notation, where 𝐵(𝑥) is not constant but depends on 𝑥. (Contributed by NM, 29-Dec-2014.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)⟶𝐷) | ||
Theorem | fmpt2 5855* | Functionality, domain and range of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 ↔ 𝐹:(𝐴 × 𝐵)⟶𝐷) | ||
Theorem | fnmpt2 5856* | Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → 𝐹 Fn (𝐴 × 𝐵)) | ||
Theorem | mpt2fvex 5857* | Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ ((∀𝑥∀𝑦 𝐶 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ∧ 𝑆 ∈ 𝑋) → (𝑅𝐹𝑆) ∈ V) | ||
Theorem | fnmpt2i 5858* | Functionality and domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ 𝐶 ∈ V ⇒ ⊢ 𝐹 Fn (𝐴 × 𝐵) | ||
Theorem | dmmpt2 5859* | Domain of a class given by the "maps to" notation. (Contributed by FL, 17-May-2010.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ 𝐶 ∈ V ⇒ ⊢ dom 𝐹 = (𝐴 × 𝐵) | ||
Theorem | mpt2fvexi 5860* | Sufficient condition for an operation maps-to notation to be set-like. (Contributed by Mario Carneiro, 3-Jul-2019.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ 𝐶 ∈ V & ⊢ 𝑅 ∈ V & ⊢ 𝑆 ∈ V ⇒ ⊢ (𝑅𝐹𝑆) ∈ V | ||
Theorem | mpt2exxg 5861* | Existence of an operation class abstraction (version for dependent domains). (Contributed by Mario Carneiro, 30-Dec-2016.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ ((𝐴 ∈ 𝑅 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑆) → 𝐹 ∈ V) | ||
Theorem | mpt2exg 5862* | Existence of an operation class abstraction (special case). (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 1-Sep-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → 𝐹 ∈ V) | ||
Theorem | mpt2exga 5863* | If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 12-Sep-2011.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) | ||
Theorem | mpt2ex 5864* | If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by Mario Carneiro, 20-Dec-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V | ||
Theorem | fmpt2co 5865* | Composition of two functions. Variation of fmptco 5358 when the second function has two arguments. (Contributed by Mario Carneiro, 8-Feb-2015.) |
⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑅 ∈ 𝐶) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑅)) & ⊢ (𝜑 → 𝐺 = (𝑧 ∈ 𝐶 ↦ 𝑆)) & ⊢ (𝑧 = 𝑅 → 𝑆 = 𝑇) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑇)) | ||
Theorem | oprabco 5866* | Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 26-Sep-2015.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐷) & ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐻‘𝐶)) ⇒ ⊢ (𝐻 Fn 𝐷 → 𝐺 = (𝐻 ∘ 𝐹)) | ||
Theorem | oprab2co 5867* | Composition of operator abstractions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by David Abernethy, 23-Apr-2013.) |
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝑅) & ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝑆) & ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ ⟨𝐶, 𝐷⟩) & ⊢ 𝐺 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝐶𝑀𝐷)) ⇒ ⊢ (𝑀 Fn (𝑅 × 𝑆) → 𝐺 = (𝑀 ∘ 𝐹)) | ||
Theorem | df1st2 5868* | An alternate possible definition of the 1^{st} function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1^{st} ↾ (V × V)) | ||
Theorem | df2nd2 5869* | An alternate possible definition of the 2^{nd} function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑦} = (2^{nd} ↾ (V × V)) | ||
Theorem | 1stconst 5870 | The mapping of a restriction of the 1^{st} function to a constant function. (Contributed by NM, 14-Dec-2008.) |
⊢ (𝐵 ∈ 𝑉 → (1^{st} ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto→𝐴) | ||
Theorem | 2ndconst 5871 | The mapping of a restriction of the 2^{nd} function to a converse constant function. (Contributed by NM, 27-Mar-2008.) |
⊢ (𝐴 ∈ 𝑉 → (2^{nd} ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto→𝐵) | ||
Theorem | dfmpt2 5872* | Alternate definition for the "maps to" notation df-mpt2 5545 (although it requires that 𝐶 be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐶 ∈ V ⇒ ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩} | ||
Theorem | cnvf1olem 5873 | Lemma for cnvf1o 5874. (Contributed by Mario Carneiro, 27-Apr-2014.) |
⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^{◡}{𝐵})) → (𝐶 ∈ ^{◡}𝐴 ∧ 𝐵 = ∪ ^{◡}{𝐶})) | ||
Theorem | cnvf1o 5874* | Describe a function that maps the elements of a set to its converse bijectively. (Contributed by Mario Carneiro, 27-Apr-2014.) |
⊢ (Rel 𝐴 → (𝑥 ∈ 𝐴 ↦ ∪ ^{◡}{𝑥}):𝐴–1-1-onto→^{◡}𝐴) | ||
Theorem | f2ndf 5875 | The 2^{nd} (second member of an ordered pair) function restricted to a function 𝐹 is a function of 𝐹 into the codomain of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.) |
⊢ (𝐹:𝐴⟶𝐵 → (2^{nd} ↾ 𝐹):𝐹⟶𝐵) | ||
Theorem | fo2ndf 5876 | The 2^{nd} (second member of an ordered pair) function restricted to a function 𝐹 is a function of 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.) |
⊢ (𝐹:𝐴⟶𝐵 → (2^{nd} ↾ 𝐹):𝐹–onto→ran 𝐹) | ||
Theorem | f1o2ndf1 5877 | The 2^{nd} (second member of an ordered pair) function restricted to a one-to-one function 𝐹 is a one-to-one function of 𝐹 onto the range of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.) |
⊢ (𝐹:𝐴–1-1→𝐵 → (2^{nd} ↾ 𝐹):𝐹–1-1-onto→ran 𝐹) | ||
Theorem | algrflem 5878 | Lemma for algrf and related theorems. (Contributed by Mario Carneiro, 28-May-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐵(𝐹 ∘ 1^{st} )𝐶) = (𝐹‘𝐵) | ||
Theorem | algrflemg 5879 | Lemma for algrf and related theorems. (Contributed by Jim Kingdon, 22-Jul-2021.) |
⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(𝐹 ∘ 1^{st} )𝐶) = (𝐹‘𝐵)) | ||
Theorem | xporderlem 5880* | Lemma for lexicographical ordering theorems. (Contributed by Scott Fenton, 16-Mar-2011.) |
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1^{st} ‘𝑥)𝑅(1^{st} ‘𝑦) ∨ ((1^{st} ‘𝑥) = (1^{st} ‘𝑦) ∧ (2^{nd} ‘𝑥)𝑆(2^{nd} ‘𝑦))))} ⇒ ⊢ (⟨𝑎, 𝑏⟩𝑇⟨𝑐, 𝑑⟩ ↔ (((𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ (𝑎𝑅𝑐 ∨ (𝑎 = 𝑐 ∧ 𝑏𝑆𝑑)))) | ||
Theorem | poxp 5881* | A lexicographical ordering of two posets. (Contributed by Scott Fenton, 16-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.) |
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝑦 ∈ (𝐴 × 𝐵)) ∧ ((1^{st} ‘𝑥)𝑅(1^{st} ‘𝑦) ∨ ((1^{st} ‘𝑥) = (1^{st} ‘𝑦) ∧ (2^{nd} ‘𝑥)𝑆(2^{nd} ‘𝑦))))} ⇒ ⊢ ((𝑅 Po 𝐴 ∧ 𝑆 Po 𝐵) → 𝑇 Po (𝐴 × 𝐵)) | ||
Theorem | spc2ed 5882* | Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.) |
⊢ Ⅎ𝑥𝜒 & ⊢ Ⅎ𝑦𝜒 & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒)) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (𝜒 → ∃𝑥∃𝑦𝜓)) | ||
Theorem | cnvoprab 5883* | The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝜓 & ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜑)) & ⊢ (𝜓 → 𝑎 ∈ (V × V)) ⇒ ⊢ ^{◡}{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓} | ||
Theorem | f1od2 5884* | Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌)) & ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽)))) ⇒ ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷) | ||
The following theorems are about maps-to operations (see df-mpt2 5545) where the first argument is a pair and the base set of the second argument is the first component of the first argument, in short "x-maps-to operations". For labels, the abbreviations "mpt2x" are used (since "x" usually denotes the first argument). This is in line with the currently used conventions for such cases (see cbvmpt2x 5610, ovmpt2x 5657 and fmpt2x 5854). However, there is a proposal by Norman Megill to use the abbreviation "mpo" or "mpto" instead of "mpt2" (see beginning of set.mm). If this proposal will be realized, the labels in the following should also be adapted. If the first argument is an ordered pair, as in the following, the abbreviation is extended to "mpt2xop", and the maps-to operations are called "x-op maps-to operations" for short. | ||
Theorem | mpt2xopn0yelv 5885* | If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.) |
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1^{st} ‘𝑥) ↦ 𝐶) ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) → 𝐾 ∈ 𝑉)) | ||
Theorem | mpt2xopoveq 5886* | Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens, 11-Oct-2017.) |
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1^{st} ‘𝑥) ↦ {𝑛 ∈ (1^{st} ‘𝑥) ∣ 𝜑}) ⇒ ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (⟨𝑉, 𝑊⟩𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦]𝜑}) | ||
Theorem | mpt2xopovel 5887* | Element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument. (Contributed by Alexander van der Vekens and Mario Carneiro, 10-Oct-2017.) |
⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1^{st} ‘𝑥) ↦ {𝑛 ∈ (1^{st} ‘𝑥) ∣ 𝜑}) ⇒ ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (⟨𝑉, 𝑊⟩𝐹𝐾) ↔ (𝐾 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉 ∧ [⟨𝑉, 𝑊⟩ / 𝑥][𝐾 / 𝑦][𝑁 / 𝑛]𝜑))) | ||
Theorem | sprmpt2 5888* | The extension of a binary relation which is the value of an operation given in maps-to notation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) |
⊢ 𝑀 = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣𝑊𝑒)𝑝 ∧ 𝜒)}) & ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜒 ↔ 𝜓)) & ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑓(𝑉𝑊𝐸)𝑝 → 𝜃)) & ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ 𝜃} ∈ V) ⇒ ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉𝑀𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝 ∧ 𝜓)}) | ||
Theorem | isprmpt2 5889* | Properties of a pair in an extended binary relation. (Contributed by Alexander van der Vekens, 30-Oct-2017.) |
⊢ (𝜑 → 𝑀 = {⟨𝑓, 𝑝⟩ ∣ (𝑓𝑊𝑝 ∧ 𝜓)}) & ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → ((𝐹 ∈ 𝑋 ∧ 𝑃 ∈ 𝑌) → (𝐹𝑀𝑃 ↔ (𝐹𝑊𝑃 ∧ 𝜒)))) | ||
Syntax | ctpos 5890 | The transposition of a function. |
class tpos 𝐹 | ||
Definition | df-tpos 5891* | Define the transposition of a function, which is a function 𝐺 = tpos 𝐹 satisfying 𝐺(𝑥, 𝑦) = 𝐹(𝑦, 𝑥). (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ tpos 𝐹 = (𝐹 ∘ (𝑥 ∈ (^{◡}dom 𝐹 ∪ {∅}) ↦ ∪ ^{◡}{𝑥})) | ||
Theorem | tposss 5892 | Subset theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺) | ||
Theorem | tposeq 5893 | Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺) | ||
Theorem | tposeqd 5894 | Equality theorem for transposition. (Contributed by Mario Carneiro, 7-Jan-2017.) |
⊢ (𝜑 → 𝐹 = 𝐺) ⇒ ⊢ (𝜑 → tpos 𝐹 = tpos 𝐺) | ||
Theorem | tposssxp 5895 | The transposition is a subset of a cross product. (Contributed by Mario Carneiro, 12-Jan-2017.) |
⊢ tpos 𝐹 ⊆ ((^{◡}dom 𝐹 ∪ {∅}) × ran 𝐹) | ||
Theorem | reltpos 5896 | The transposition is a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ Rel tpos 𝐹 | ||
Theorem | brtpos2 5897 | Value of the transposition at a pair ⟨𝐴, 𝐵⟩. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐵 ∈ 𝑉 → (𝐴tpos 𝐹𝐵 ↔ (𝐴 ∈ (^{◡}dom 𝐹 ∪ {∅}) ∧ ∪ ^{◡}{𝐴}𝐹𝐵))) | ||
Theorem | brtpos0 5898 | The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴)) | ||
Theorem | reldmtpos 5899 | Necessary and sufficient condition for dom tpos 𝐹 to be a relation. (Contributed by Mario Carneiro, 10-Sep-2015.) |
⊢ (Rel dom tpos 𝐹 ↔ ¬ ∅ ∈ dom 𝐹) | ||
Theorem | brtposg 5900 | The transposition swaps arguments of a three-parameter relation. (Contributed by Jim Kingdon, 31-Jan-2019.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨𝐴, 𝐵⟩tpos 𝐹𝐶 ↔ ⟨𝐵, 𝐴⟩𝐹𝐶)) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |