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Type | Label | Description |
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Statement | ||
Theorem | oprabrexex2 8001* | Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) |
⊢ 𝐴 ∈ V & ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ∈ V ⇒ ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ∃𝑤 ∈ 𝐴 𝜑} ∈ V | ||
Theorem | ab2rexex 8002* | Existence of a class abstraction of existentially restricted sets. Variables 𝑥 and 𝑦 are normally free-variable parameters in the class expression substituted for 𝐶, which can be thought of as 𝐶(𝑥, 𝑦). See comments for abrexex 7985. (Contributed by NM, 20-Sep-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} ∈ V | ||
Theorem | ab2rexex2 8003* | Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥, 𝑦, and 𝑧. Compare abrexex2 7992. (Contributed by NM, 20-Sep-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ {𝑧 ∣ 𝜑} ∈ V ⇒ ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑} ∈ V | ||
Theorem | xpexgALT 8004 | Alternate proof of xpexg 7768 requiring Replacement (ax-rep 5284) but not Power Set (ax-pow 5370). (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V) | ||
Theorem | offval3 8005* | General value of (𝐹 ∘f 𝑅𝐺) with no assumptions on functionality of 𝐹 and 𝐺. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) | ||
Theorem | offres 8006 | Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘f 𝑅𝐺) ↾ 𝐷) = ((𝐹 ↾ 𝐷) ∘f 𝑅(𝐺 ↾ 𝐷))) | ||
Theorem | ofmres 8007* | Equivalent expressions for a restriction of the function operation map. Unlike ∘f 𝑅 which is a proper class, ( ∘f 𝑅 ↾ (𝐴 × 𝐵)) can be a set by ofmresex 8008, allowing it to be used as a function or structure argument. By ofmresval 7712, the restricted operation map values are the same as the original values, allowing theorems for ∘f 𝑅 to be reused. (Contributed by NM, 20-Oct-2014.) |
⊢ ( ∘f 𝑅 ↾ (𝐴 × 𝐵)) = (𝑓 ∈ 𝐴, 𝑔 ∈ 𝐵 ↦ (𝑓 ∘f 𝑅𝑔)) | ||
Theorem | ofmresex 8008 | Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → ( ∘f 𝑅 ↾ (𝐴 × 𝐵)) ∈ V) | ||
Theorem | mptcnfimad 8009* | The converse of a mapping of subsets to their image of a bijection. (Contributed by AV, 23-Apr-2025.) |
⊢ 𝑀 = (𝑥 ∈ 𝐴 ↦ (𝐹 “ 𝑥)) & ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝑊) & ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝑉) & ⊢ (𝜑 → ran 𝑀 ⊆ 𝒫 𝑊) & ⊢ (𝜑 → 𝑉 ∈ 𝑈) ⇒ ⊢ (𝜑 → ◡𝑀 = (𝑦 ∈ ran 𝑀 ↦ (◡𝐹 “ 𝑦))) | ||
Syntax | c1st 8010 | Extend the definition of a class to include the first member an ordered pair function. |
class 1st | ||
Syntax | c2nd 8011 | Extend the definition of a class to include the second member an ordered pair function. |
class 2nd | ||
Definition | df-1st 8012 | Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 8020 proves that it does this. For example, (1st ‘〈3, 4〉) = 3. Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 6246 and op1stb 5481). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.) |
⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥}) | ||
Definition | df-2nd 8013 | Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 8021 proves that it does this. For example, (2nd ‘〈3, 4〉) = 4. Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 6249 and op2ndb 6248). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.) |
⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | ||
Theorem | 1stval 8014 | The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (1st ‘𝐴) = ∪ dom {𝐴} | ||
Theorem | 2ndval 8015 | The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (2nd ‘𝐴) = ∪ ran {𝐴} | ||
Theorem | 1stnpr 8016 | Value of the first-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.) |
⊢ (¬ 𝐴 ∈ (V × V) → (1st ‘𝐴) = ∅) | ||
Theorem | 2ndnpr 8017 | Value of the second-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.) |
⊢ (¬ 𝐴 ∈ (V × V) → (2nd ‘𝐴) = ∅) | ||
Theorem | 1st0 8018 | The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.) |
⊢ (1st ‘∅) = ∅ | ||
Theorem | 2nd0 8019 | The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.) |
⊢ (2nd ‘∅) = ∅ | ||
Theorem | op1st 8020 | Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (1st ‘〈𝐴, 𝐵〉) = 𝐴 | ||
Theorem | op2nd 8021 | Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (2nd ‘〈𝐴, 𝐵〉) = 𝐵 | ||
Theorem | op1std 8022 | Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐶 = 〈𝐴, 𝐵〉 → (1st ‘𝐶) = 𝐴) | ||
Theorem | op2ndd 8023 | Extract the second member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐶 = 〈𝐴, 𝐵〉 → (2nd ‘𝐶) = 𝐵) | ||
Theorem | op1stg 8024 | Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘〈𝐴, 𝐵〉) = 𝐴) | ||
Theorem | op2ndg 8025 | Extract the second member of an ordered pair. (Contributed by NM, 19-Jul-2005.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (2nd ‘〈𝐴, 𝐵〉) = 𝐵) | ||
Theorem | ot1stg 8026 | 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 8026, ot2ndg 8027, ot3rdg 8028.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1st ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐴) | ||
Theorem | ot2ndg 8027 | Extract the second member of an ordered triple. (See ot1stg 8026 comment.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (2nd ‘(1st ‘〈𝐴, 𝐵, 𝐶〉)) = 𝐵) | ||
Theorem | ot3rdg 8028 | Extract the third member of an ordered triple. (See ot1stg 8026 comment.) (Contributed by NM, 3-Apr-2015.) |
⊢ (𝐶 ∈ 𝑉 → (2nd ‘〈𝐴, 𝐵, 𝐶〉) = 𝐶) | ||
Theorem | 1stval2 8029 | 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) → (1st ‘𝐴) = ∩ ∩ 𝐴) | ||
Theorem | 2ndval2 8030 | 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) → (2nd ‘𝐴) = ∩ ∩ ∩ ◡{𝐴}) | ||
Theorem | oteqimp 8031 | The components of an ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.) |
⊢ (𝑇 = 〈𝐴, 𝐵, 𝐶〉 → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ((1st ‘(1st ‘𝑇)) = 𝐴 ∧ (2nd ‘(1st ‘𝑇)) = 𝐵 ∧ (2nd ‘𝑇) = 𝐶))) | ||
Theorem | fo1st 8032 | The 1st function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ 1st :V–onto→V | ||
Theorem | fo2nd 8033 | The 2nd function maps the universe onto the universe. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ 2nd :V–onto→V | ||
Theorem | br1steqg 8034 | Uniqueness condition for the binary relation 1st. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on 𝐶. (Revised by Peter Mazsa, 17-Jan-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉1st 𝐶 ↔ 𝐶 = 𝐴)) | ||
Theorem | br2ndeqg 8035 | Uniqueness condition for the binary relation 2nd. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on 𝐶. (Revised by Peter Mazsa, 17-Jan-2022.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉2nd 𝐶 ↔ 𝐶 = 𝐵)) | ||
Theorem | f1stres 8036 | Mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 | ||
Theorem | f2ndres 8037 | Mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.) |
⊢ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 | ||
Theorem | fo1stres 8038 | Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.) |
⊢ (𝐵 ≠ ∅ → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐴) | ||
Theorem | fo2ndres 8039 | Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.) |
⊢ (𝐴 ≠ ∅ → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto→𝐵) | ||
Theorem | 1st2val 8040* | Value of an alternate definition of the 1st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑥}‘𝐴) = (1st ‘𝐴) | ||
Theorem | 2nd2val 8041* | Value of an alternate definition of the 2nd function. (Contributed by NM, 10-Aug-2006.) (Revised by Mario Carneiro, 30-Dec-2014.) |
⊢ ({〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝑧 = 𝑦}‘𝐴) = (2nd ‘𝐴) | ||
Theorem | 1stcof 8042 | Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.) |
⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st ∘ 𝐹):𝐴⟶𝐵) | ||
Theorem | 2ndcof 8043 | Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.) |
⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹):𝐴⟶𝐶) | ||
Theorem | xp1st 8044 | Location of the first element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → (1st ‘𝐴) ∈ 𝐵) | ||
Theorem | xp2nd 8045 | Location of the second element of a Cartesian product. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → (2nd ‘𝐴) ∈ 𝐶) | ||
Theorem | elxp6 8046 | Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7944. (Contributed by NM, 9-Oct-2004.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) | ||
Theorem | elxp7 8047 | Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7944. (Contributed by NM, 19-Aug-2006.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))) | ||
Theorem | eqopi 8048 | Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.) |
⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) → 𝐴 = 〈𝐵, 𝐶〉) | ||
Theorem | xp2 8049* | Representation of Cartesian product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.) |
⊢ (𝐴 × 𝐵) = {𝑥 ∈ (V × V) ∣ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ 𝐵)} | ||
Theorem | unielxp 8050 | The membership relation for a Cartesian product is inherited by union. (Contributed by NM, 16-Sep-2006.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ 𝐴 ∈ ∪ (𝐵 × 𝐶)) | ||
Theorem | 1st2nd2 8051 | Reconstruction of a member of a Cartesian product in terms of its ordered pair components. (Contributed by NM, 20-Oct-2013.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | ||
Theorem | 1st2ndb 8052 | Reconstruction of an ordered pair in terms of its components. (Contributed by NM, 25-Feb-2014.) |
⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | ||
Theorem | xpopth 8053 | An ordered pair theorem for members of Cartesian products. (Contributed by NM, 20-Jun-2007.) |
⊢ ((𝐴 ∈ (𝐶 × 𝐷) ∧ 𝐵 ∈ (𝑅 × 𝑆)) → (((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)) ↔ 𝐴 = 𝐵)) | ||
Theorem | eqop 8054 | 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〉 ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) | ||
Theorem | eqop2 8055 | 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) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))) | ||
Theorem | op1steq 8056* | 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) → ((1st ‘𝐴) = 𝐵 ↔ ∃𝑥 𝐴 = 〈𝐵, 𝑥〉)) | ||
Theorem | opreuopreu 8057* | There is a unique ordered pair fulfilling a wff iff its components fulfil a corresponding wff. (Contributed by AV, 2-Jul-2023.) |
⊢ ((𝑎 = (1st ‘𝑝) ∧ 𝑏 = (2nd ‘𝑝)) → (𝜓 ↔ 𝜑)) ⇒ ⊢ (∃!𝑝 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃!𝑝 ∈ (𝐴 × 𝐵)∃𝑎∃𝑏(𝑝 = 〈𝑎, 𝑏〉 ∧ 𝜓)) | ||
Theorem | el2xptp 8058* | A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.) |
⊢ (𝐴 ∈ ((𝐵 × 𝐶) × 𝐷) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 ∃𝑧 ∈ 𝐷 𝐴 = 〈𝑥, 𝑦, 𝑧〉) | ||
Theorem | el2xptp0 8059 | A member of a nested Cartesian product is an ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.) |
⊢ ((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → ((𝐴 ∈ ((𝑈 × 𝑉) × 𝑊) ∧ ((1st ‘(1st ‘𝐴)) = 𝑋 ∧ (2nd ‘(1st ‘𝐴)) = 𝑌 ∧ (2nd ‘𝐴) = 𝑍)) ↔ 𝐴 = 〈𝑋, 𝑌, 𝑍〉)) | ||
Theorem | el2xpss 8060* | Version of elrel 5810 for triple Cartesian products. (Contributed by Scott Fenton, 1-Feb-2025.) |
⊢ ((𝐴 ∈ 𝑅 ∧ 𝑅 ⊆ ((𝐵 × 𝐶) × 𝐷)) → ∃𝑥∃𝑦∃𝑧 𝐴 = 〈𝑥, 𝑦, 𝑧〉) | ||
Theorem | 2nd1st 8061 | Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.) |
⊢ (𝐴 ∈ (𝐵 × 𝐶) → ∪ ◡{𝐴} = 〈(2nd ‘𝐴), (1st ‘𝐴)〉) | ||
Theorem | 1st2nd 8062 | 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 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | ||
Theorem | 1stdm 8063 | 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) → (1st ‘𝐴) ∈ dom 𝑅) | ||
Theorem | 2ndrn 8064 | 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) → (2nd ‘𝐴) ∈ ran 𝑅) | ||
Theorem | 1st2ndbr 8065 | 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) → (1st ‘𝐴)𝐵(2nd ‘𝐴)) | ||
Theorem | releldm2 8066* | Two ways of expressing membership in the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
⊢ (Rel 𝐴 → (𝐵 ∈ dom 𝐴 ↔ ∃𝑥 ∈ 𝐴 (1st ‘𝑥) = 𝐵)) | ||
Theorem | reldm 8067* | An expression for the domain of a relation. (Contributed by NM, 22-Sep-2013.) |
⊢ (Rel 𝐴 → dom 𝐴 = ran (𝑥 ∈ 𝐴 ↦ (1st ‘𝑥))) | ||
Theorem | releldmdifi 8068* | One way of expressing membership in the difference of domains of two nested relations. (Contributed by AV, 26-Oct-2023.) |
⊢ ((Rel 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) → ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶)) | ||
Theorem | funfv1st2nd 8069 | The function value for the first component of an ordered pair is the second component of the ordered pair. (Contributed by AV, 17-Oct-2023.) |
⊢ ((Fun 𝐹 ∧ 𝑋 ∈ 𝐹) → (𝐹‘(1st ‘𝑋)) = (2nd ‘𝑋)) | ||
Theorem | funelss 8070 | If the first component of an element of a function is in the domain of a subset of the function, the element is a member of this subset. (Contributed by AV, 27-Oct-2023.) |
⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑋 ∈ 𝐴) → ((1st ‘𝑋) ∈ dom 𝐵 → 𝑋 ∈ 𝐵)) | ||
Theorem | funeldmdif 8071* | Two ways of expressing membership in the difference of domains of two nested functions. (Contributed by AV, 27-Oct-2023.) |
⊢ ((Fun 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (dom 𝐴 ∖ dom 𝐵) ↔ ∃𝑥 ∈ (𝐴 ∖ 𝐵)(1st ‘𝑥) = 𝐶)) | ||
Theorem | sbcopeq1a 8072 | Equality theorem for substitution of a class for an ordered pair (analogue of sbceq1a 3801 that avoids the existential quantifiers of copsexg 5501). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 = 〈𝑥, 𝑦〉 → ([(1st ‘𝐴) / 𝑥][(2nd ‘𝐴) / 𝑦]𝜑 ↔ 𝜑)) | ||
Theorem | csbopeq1a 8073 | Equality theorem for substitution of a class 𝐴 for an ordered pair 〈𝑥, 𝑦〉 in 𝐵 (analogue of csbeq1a 3921). (Contributed by NM, 19-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐴 = 〈𝑥, 𝑦〉 → ⦋(1st ‘𝐴) / 𝑥⦌⦋(2nd ‘𝐴) / 𝑦⦌𝐵 = 𝐵) | ||
Theorem | sbcoteq1a 8074 | Equality theorem for substitution of a class for an ordered triple. (Contributed by Scott Fenton, 22-Aug-2024.) |
⊢ (𝐴 = 〈𝑥, 𝑦, 𝑧〉 → ([(1st ‘(1st ‘𝐴)) / 𝑥][(2nd ‘(1st ‘𝐴)) / 𝑦][(2nd ‘𝐴) / 𝑧]𝜑 ↔ 𝜑)) | ||
Theorem | dfopab2 8075* | 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) ∣ [(1st ‘𝑧) / 𝑥][(2nd ‘𝑧) / 𝑦]𝜑} | ||
Theorem | dfoprab3s 8076* | 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) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)} | ||
Theorem | dfoprab3 8077* | 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 8078* | 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 8079* | Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 20-Dec-2008.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (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 | opabex2 8080* | Condition for an operation to be a set. (Contributed by Thierry Arnoux, 25-Jun-2019.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) & ⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴) & ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵) ⇒ ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜓} ∈ V) | ||
Theorem | opabn1stprc 8081* | An ordered-pair class abstraction which does not depend on the first abstraction variable is a proper class. There must be, however, at least one set which satisfies the restricting wff. (Contributed by AV, 27-Dec-2020.) |
⊢ (∃𝑦𝜑 → {〈𝑥, 𝑦〉 ∣ 𝜑} ∉ V) | ||
Theorem | opiota 8082* | The property of a uniquely specified ordered pair. The proof uses properties of the ℩ description binder. (Contributed by Mario Carneiro, 21-May-2015.) |
⊢ 𝐼 = (℩𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑)) & ⊢ 𝑋 = (1st ‘𝐼) & ⊢ 𝑌 = (2nd ‘𝐼) & ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐷 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∃!𝑧∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑧 = 〈𝑥, 𝑦〉 ∧ 𝜑) → ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵 ∧ 𝜒) ↔ (𝐶 = 𝑋 ∧ 𝐷 = 𝑌))) | ||
Theorem | cnvoprab 8083* | The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.) (Proof shortened by Thierry Arnoux, 20-Feb-2022.) |
⊢ (𝑎 = 〈𝑥, 𝑦〉 → (𝜓 ↔ 𝜑)) & ⊢ (𝜓 → 𝑎 ∈ (V × V)) ⇒ ⊢ ◡{〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} = {〈𝑧, 𝑎〉 ∣ 𝜓} | ||
Theorem | dfxp3 8084* | Define the Cartesian product of three classes. Compare df-xp 5694. (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 8085* | A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.) |
⊢ (𝑥 = (1st ‘𝐴) → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = (2nd ‘𝐴) → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → 𝜒) | ||
Theorem | eloprabi 8086* | 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 = (1st ‘(1st ‘𝐴)) → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = (2nd ‘(1st ‘𝐴)) → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = (2nd ‘𝐴) → (𝜒 ↔ 𝜃)) ⇒ ⊢ (𝐴 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} → 𝜃) | ||
Theorem | mpomptsx 8087* | 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) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶) | ||
Theorem | mpompts 8088* | 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) ↦ ⦋(1st ‘𝑧) / 𝑥⦌⦋(2nd ‘𝑧) / 𝑦⦌𝐶) | ||
Theorem | dmmpossx 8089* | 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 | fmpox 8090* | 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 | fmpo 8091* | 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 | fnmpo 8092* | 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 | fnmpoi 8093* | 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 | dmmpo 8094* | 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 | ovmpoelrn 8095* | An operation's value belongs to its range. (Contributed by AV, 27-Jan-2020.) |
⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑀 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) ∈ 𝑀) | ||
Theorem | dmmpoga 8096* | Domain of an operation given by the maps-to notation, closed form of dmmpo 8094. (Contributed by Alexander van der Vekens, 10-Feb-2019.) (Proof shortened by Lammen, 29-May-2024.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵)) | ||
Theorem | dmmpog 8097* | Domain of an operation given by the maps-to notation, closed form of dmmpo 8094. Caution: This theorem is only valid in the very special case where the value of the mapping is a constant! (Contributed by Alexander van der Vekens, 1-Jun-2017.) (Proof shortened by AV, 10-Feb-2019.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (𝐶 ∈ 𝑉 → dom 𝐹 = (𝐴 × 𝐵)) | ||
Theorem | mpoexxg 8098* | 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 | mpoexg 8099* | 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 | mpoexga 8100* | If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by NM, 12-Sep-2011.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) |
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