![]() |
Metamath
Proof Explorer Theorem List (p. 76 of 480) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | ![]() (1-30438) |
![]() (30439-31961) |
![]() (31962-47939) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cbvoprab12v 7501* | Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) |
⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑣) → (𝜑 ↔ 𝜓)) ⇒ ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑤, 𝑣⟩, 𝑧⟩ ∣ 𝜓} | ||
Theorem | cbvoprab3 7502* | Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 22-Aug-2013.) |
⊢ Ⅎ𝑤𝜑 & ⊢ Ⅎ𝑧𝜓 & ⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓} | ||
Theorem | cbvoprab3v 7503* | Rule used to change the third bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 8-Oct-2004.) (Revised by David Abernethy, 19-Jun-2012.) |
⊢ (𝑧 = 𝑤 → (𝜑 ↔ 𝜓)) ⇒ ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ 𝜓} | ||
Theorem | cbvmpox 7504* | Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 7505 allows 𝐵 to be a function of 𝑥. (Contributed by NM, 29-Dec-2014.) |
⊢ Ⅎ𝑧𝐵 & ⊢ Ⅎ𝑥𝐷 & ⊢ Ⅎ𝑧𝐶 & ⊢ Ⅎ𝑤𝐶 & ⊢ Ⅎ𝑥𝐸 & ⊢ Ⅎ𝑦𝐸 & ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐷) & ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐸) ⇒ ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐷 ↦ 𝐸) | ||
Theorem | cbvmpo 7505* | Rule to change the bound variable in a maps-to function, using implicit substitution. (Contributed by NM, 17-Dec-2013.) |
⊢ Ⅎ𝑧𝐶 & ⊢ Ⅎ𝑤𝐶 & ⊢ Ⅎ𝑥𝐷 & ⊢ Ⅎ𝑦𝐷 & ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷) ⇒ ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) | ||
Theorem | cbvmpov 7506* | Rule to change the bound variable in a maps-to function, using implicit substitution. With a longer proof analogous to cbvmpt 5258, some distinct variable requirements could be eliminated. (Contributed by NM, 11-Jun-2013.) |
⊢ (𝑥 = 𝑧 → 𝐶 = 𝐸) & ⊢ (𝑦 = 𝑤 → 𝐸 = 𝐷) ⇒ ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑧 ∈ 𝐴, 𝑤 ∈ 𝐵 ↦ 𝐷) | ||
Theorem | elimdelov 7507 | Eliminate a hypothesis which is a predicate expressing membership in the result of an operator (deduction version). (Contributed by Paul Chapman, 25-Mar-2008.) |
⊢ (𝜑 → 𝐶 ∈ (𝐴𝐹𝐵)) & ⊢ 𝑍 ∈ (𝑋𝐹𝑌) ⇒ ⊢ if(𝜑, 𝐶, 𝑍) ∈ (if(𝜑, 𝐴, 𝑋)𝐹if(𝜑, 𝐵, 𝑌)) | ||
Theorem | ovif 7508 | Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
⊢ (if(𝜑, 𝐴, 𝐵)𝐹𝐶) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐶)) | ||
Theorem | ovif2 7509 | Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 1-Oct-2018.) |
⊢ (𝐴𝐹if(𝜑, 𝐵, 𝐶)) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐹𝐶)) | ||
Theorem | ovif12 7510 | Move a conditional outside of an operation. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
⊢ (if(𝜑, 𝐴, 𝐵)𝐹if(𝜑, 𝐶, 𝐷)) = if(𝜑, (𝐴𝐹𝐶), (𝐵𝐹𝐷)) | ||
Theorem | ifov 7511 | Move a conditional outside of an operation. (Contributed by AV, 11-Nov-2019.) |
⊢ (𝐴if(𝜑, 𝐹, 𝐺)𝐵) = if(𝜑, (𝐴𝐹𝐵), (𝐴𝐺𝐵)) | ||
Theorem | dmoprab 7512* | The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝜑} | ||
Theorem | dmoprabss 7513* | The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.) |
⊢ dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ⊆ (𝐴 × 𝐵) | ||
Theorem | rnoprab 7514* | The range of an operation class abstraction. (Contributed by NM, 30-Aug-2004.) (Revised by David Abernethy, 19-Apr-2013.) |
⊢ ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑧 ∣ ∃𝑥∃𝑦𝜑} | ||
Theorem | rnoprab2 7515* | The range of a restricted operation class abstraction. (Contributed by Scott Fenton, 21-Mar-2012.) |
⊢ ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑} | ||
Theorem | reldmoprab 7516* | The domain of an operation class abstraction is a relation. (Contributed by NM, 17-Mar-1995.) |
⊢ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} | ||
Theorem | oprabss 7517* | Structure of an operation class abstraction. (Contributed by NM, 28-Nov-2006.) |
⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ ((V × V) × V) | ||
Theorem | eloprabga 7518* | The law of concretion for operation class abstraction. Compare elopab 5526. (Contributed by NM, 14-Sep-1999.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.) Avoid ax-10 2135, ax-11 2152. (Revised by Wolf Lammen, 15-Oct-2024.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓)) | ||
Theorem | eloprabgaOLD 7519* | Obsolete version of eloprabga 7518 as of 15-Oct-2024. (Contributed by NM, 14-Sep-1999.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓)) | ||
Theorem | eloprabg 7520* | The law of concretion for operation class abstraction. Compare elopab 5526. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜃)) | ||
Theorem | ssoprab2i 7521* | Inference of operation class abstraction subclass from implication. (Contributed by NM, 11-Nov-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⊆ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} | ||
Theorem | mpov 7522* | Operation with universal domain in maps-to notation. (Contributed by NM, 16-Aug-2013.) |
⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝐶} | ||
Theorem | mpomptx 7523* | Express a two-argument function as a one-argument function, or vice-versa. In this version 𝐵(𝑥) is not assumed to be constant w.r.t 𝑥. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷) ⇒ ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) | ||
Theorem | mpompt 7524* | Express a two-argument function as a one-argument function, or vice-versa. (Contributed by Mario Carneiro, 17-Dec-2013.) (Revised by Mario Carneiro, 29-Dec-2014.) |
⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷) ⇒ ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) | ||
Theorem | mpodifsnif 7525 | A mapping with two arguments with the first argument from a difference set with a singleton and a conditional as result. (Contributed by AV, 13-Feb-2019.) |
⊢ (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ (𝐴 ∖ {𝑋}), 𝑗 ∈ 𝐵 ↦ 𝐷) | ||
Theorem | mposnif 7526 | A mapping with two arguments with the first argument from a singleton and a conditional as result. (Contributed by AV, 14-Feb-2019.) |
⊢ (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ if(𝑖 = 𝑋, 𝐶, 𝐷)) = (𝑖 ∈ {𝑋}, 𝑗 ∈ 𝐵 ↦ 𝐶) | ||
Theorem | fconstmpo 7527* | Representation of a constant operation using the mapping operation. (Contributed by SO, 11-Jul-2018.) |
⊢ ((𝐴 × 𝐵) × {𝐶}) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | ||
Theorem | resoprab 7528* | Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.) |
⊢ ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↾ (𝐴 × 𝐵)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} | ||
Theorem | resoprab2 7529* | Restriction of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ({⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)} ↾ (𝐶 × 𝐷)) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)}) | ||
Theorem | resmpo 7530* | Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.) |
⊢ ((𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) ↾ (𝐶 × 𝐷)) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸)) | ||
Theorem | funoprabg 7531* | "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 28-Aug-2007.) |
⊢ (∀𝑥∀𝑦∃*𝑧𝜑 → Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}) | ||
Theorem | funoprab 7532* | "At most one" is a sufficient condition for an operation class abstraction to be a function. (Contributed by NM, 17-Mar-1995.) |
⊢ ∃*𝑧𝜑 ⇒ ⊢ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} | ||
Theorem | fnoprabg 7533* | Functionality and domain of an operation class abstraction. (Contributed by NM, 28-Aug-2007.) |
⊢ (∀𝑥∀𝑦(𝜑 → ∃!𝑧𝜓) → {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑}) | ||
Theorem | mpofun 7534* | The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.) (Proof shortened by SN, 23-Jul-2024.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ Fun 𝐹 | ||
Theorem | mpofunOLD 7535* | Obsolete version of mpofun 7534 as of 23-Jul-2024. (Contributed by Scott Fenton, 21-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ Fun 𝐹 | ||
Theorem | fnoprab 7536* | Functionality and domain of an operation class abstraction. (Contributed by NM, 15-May-1995.) |
⊢ (𝜑 → ∃!𝑧𝜓) ⇒ ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (𝜑 ∧ 𝜓)} Fn {⟨𝑥, 𝑦⟩ ∣ 𝜑} | ||
Theorem | ffnov 7537* | An operation maps to a class to which all values belong. (Contributed by NM, 7-Feb-2004.) |
⊢ (𝐹:(𝐴 × 𝐵)⟶𝐶 ↔ (𝐹 Fn (𝐴 × 𝐵) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶)) | ||
Theorem | fovcld 7538 | Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Revised by Thierry Arnoux, 17-Feb-2017.) |
⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) | ||
Theorem | fovcl 7539 | Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Proof shortened by AV, 9-Mar-2025.) |
⊢ 𝐹:(𝑅 × 𝑆)⟶𝐶 ⇒ ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) | ||
Theorem | eqfnov 7540* | Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.) |
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐶 × 𝐷)) → (𝐹 = 𝐺 ↔ ((𝐴 × 𝐵) = (𝐶 × 𝐷) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦)))) | ||
Theorem | eqfnov2 7541* | Two operators with the same domain are equal iff their values at each point in the domain are equal. (Contributed by Jeff Madsen, 7-Jun-2010.) |
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐺 Fn (𝐴 × 𝐵)) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) = (𝑥𝐺𝑦))) | ||
Theorem | fnov 7542* | Representation of a function in terms of its values. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (𝐹 Fn (𝐴 × 𝐵) ↔ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑥𝐹𝑦))) | ||
Theorem | mpo2eqb 7543* | Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnov2 7541. (Contributed by Mario Carneiro, 4-Jan-2017.) |
⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 = 𝐷)) | ||
Theorem | rnmpo 7544* | The range of an operation given by the maps-to notation. (Contributed by FL, 20-Jun-2011.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = 𝐶} | ||
Theorem | reldmmpo 7545* | The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ Rel dom 𝐹 | ||
Theorem | elrnmpog 7546* | Membership in the range of an operation class abstraction. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶)) | ||
Theorem | elrnmpo 7547* | Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ 𝐶 ∈ V ⇒ ⊢ (𝐷 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐷 = 𝐶) | ||
Theorem | elrnmpores 7548* | Membership in the range of a restricted operation class abstraction. (Contributed by Thierry Arnoux, 25-May-2019.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ (𝐷 ∈ 𝑉 → (𝐷 ∈ ran (𝐹 ↾ 𝑅) ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝐷 = 𝐶 ∧ 𝑥𝑅𝑦))) | ||
Theorem | ralrnmpo 7549* | A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∀𝑧 ∈ ran 𝐹𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜓)) | ||
Theorem | rexrnmpo 7550* | A restricted quantifier over an image set. (Contributed by Mario Carneiro, 1-Sep-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) & ⊢ (𝑧 = 𝐶 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉 → (∃𝑧 ∈ ran 𝐹𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜓)) | ||
Theorem | ovid 7551* | The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃!𝑧𝜑) & ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} ⇒ ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ((𝑥𝐹𝑦) = 𝑧 ↔ 𝜑)) | ||
Theorem | ovidig 7552* | The value of an operation class abstraction. Compare ovidi 7553. The condition (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) is been removed. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ ∃*𝑧𝜑 & ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⇒ ⊢ (𝜑 → (𝑥𝐹𝑦) = 𝑧) | ||
Theorem | ovidi 7553* | The value of an operation class abstraction (weak version). (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃*𝑧𝜑) & ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} ⇒ ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → (𝜑 → (𝑥𝐹𝑦) = 𝑧)) | ||
Theorem | ov 7554* | The value of an operation class abstraction. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
⊢ 𝐶 ∈ V & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃!𝑧𝜑) & ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} ⇒ ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴𝐹𝐵) = 𝐶 ↔ 𝜃)) | ||
Theorem | ovigg 7555* | The value of an operation class abstraction. Compared with ovig 7556, the condition (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) is removed. (Contributed by FL, 24-Mar-2007.) (Revised by Mario Carneiro, 19-Dec-2013.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) & ⊢ ∃*𝑧𝜑 & ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → (𝐴𝐹𝐵) = 𝐶)) | ||
Theorem | ovig 7556* | The value of an operation class abstraction (weak version). (Contributed by NM, 14-Sep-1999.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) & ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → ∃*𝑧𝜑) & ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} ⇒ ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷) → (𝜓 → (𝐴𝐹𝐵) = 𝐶)) | ||
Theorem | ovmpt4g 7557* | Value of a function given by the maps-to notation. (This is the operation analogue of fvmpt2 7008.) (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 1-Sep-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝑉) → (𝑥𝐹𝑦) = 𝐶) | ||
Theorem | ovmpos 7558* | Value of a function given by the maps-to notation, expressed using explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅 ∈ 𝑉) → (𝐴𝐹𝐵) = ⦋𝐴 / 𝑥⦌⦋𝐵 / 𝑦⦌𝑅) | ||
Theorem | ov2gf 7559* | The value of an operation class abstraction. A version of ovmpog 7569 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑦𝐵 & ⊢ Ⅎ𝑥𝐺 & ⊢ Ⅎ𝑦𝑆 & ⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺) & ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆) & ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) | ||
Theorem | ovmpodxf 7560* | Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐿) & ⊢ (𝜑 → 𝑆 ∈ 𝑋) & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝑆 & ⊢ Ⅎ𝑦𝑆 ⇒ ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) | ||
Theorem | ovmpodx 7561* | Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 29-Dec-2014.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐷 = 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐿) & ⊢ (𝜑 → 𝑆 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) | ||
Theorem | ovmpod 7562* | Value of an operation given by a maps-to rule, deduction form. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) & ⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝑆 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) | ||
Theorem | ovmpox 7563* | The value of an operation class abstraction. Variant of ovmpoga 7564 which does not require 𝐷 and 𝑥 to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) & ⊢ (𝑥 = 𝐴 → 𝐷 = 𝐿) & ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐿 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) | ||
Theorem | ovmpoga 7564* | Value of an operation given by a maps-to rule. (Contributed by Mario Carneiro, 19-Dec-2013.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) & ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) | ||
Theorem | ovmpoa 7565* | Value of an operation given by a maps-to rule. (Contributed by NM, 19-Dec-2013.) |
⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) & ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) & ⊢ 𝑆 ∈ V ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝐹𝐵) = 𝑆) | ||
Theorem | ovmpodf 7566* | Alternate deduction version of ovmpo 7570, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅 → 𝜓)) & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑦𝐹 & ⊢ Ⅎ𝑦𝜓 ⇒ ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)) | ||
Theorem | ovmpodv 7567* | Alternate deduction version of ovmpo 7570, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅 → 𝜓)) ⇒ ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → 𝜓)) | ||
Theorem | ovmpodv2 7568* | Alternate deduction version of ovmpo 7570, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) ⇒ ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → (𝐴𝐹𝐵) = 𝑆)) | ||
Theorem | ovmpog 7569* | Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.) |
⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺) & ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆) & ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) | ||
Theorem | ovmpo 7570* | Value of an operation given by a maps-to rule. Special case. (Contributed by NM, 16-May-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
⊢ (𝑥 = 𝐴 → 𝑅 = 𝐺) & ⊢ (𝑦 = 𝐵 → 𝐺 = 𝑆) & ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) & ⊢ 𝑆 ∈ V ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝐹𝐵) = 𝑆) | ||
Theorem | fvmpopr2d 7571* | Value of an operation given by maps-to notation. (Contributed by Rohan Ridenour, 14-May-2024.) |
⊢ (𝜑 → 𝐹 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ 𝐶)) & ⊢ (𝜑 → 𝑃 = ⟨𝑎, 𝑏⟩) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐶 ∈ 𝑉) ⇒ ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘𝑃) = 𝐶) | ||
Theorem | ov3 7572* | The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.) |
⊢ 𝑆 ∈ V & ⊢ (((𝑤 = 𝐴 ∧ 𝑣 = 𝐵) ∧ (𝑢 = 𝐶 ∧ 𝑓 = 𝐷)) → 𝑅 = 𝑆) & ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ (𝐻 × 𝐻) ∧ 𝑦 ∈ (𝐻 × 𝐻)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = ⟨𝑤, 𝑣⟩ ∧ 𝑦 = ⟨𝑢, 𝑓⟩) ∧ 𝑧 = 𝑅))} ⇒ ⊢ (((𝐴 ∈ 𝐻 ∧ 𝐵 ∈ 𝐻) ∧ (𝐶 ∈ 𝐻 ∧ 𝐷 ∈ 𝐻)) → (⟨𝐴, 𝐵⟩𝐹⟨𝐶, 𝐷⟩) = 𝑆) | ||
Theorem | ov6g 7573* | The value of an operation class abstraction. Special case. (Contributed by NM, 13-Nov-2006.) |
⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ → 𝑅 = 𝑆) & ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝐶 ∧ 𝑧 = 𝑅)} ⇒ ⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐻 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐶) ∧ 𝑆 ∈ 𝐽) → (𝐴𝐹𝐵) = 𝑆) | ||
Theorem | ovg 7574* | The value of an operation class abstraction. (Contributed by Jeff Madsen, 10-Jun-2010.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ ((𝜏 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆)) → ∃!𝑧𝜑) & ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) ∧ 𝜑)} ⇒ ⊢ ((𝜏 ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝐷)) → ((𝐴𝐹𝐵) = 𝐶 ↔ 𝜃)) | ||
Theorem | ovres 7575 | The value of a restricted operation. (Contributed by FL, 10-Nov-2006.) |
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴(𝐹 ↾ (𝐶 × 𝐷))𝐵) = (𝐴𝐹𝐵)) | ||
Theorem | ovresd 7576 | Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐵 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵)) | ||
Theorem | oprres 7577* | The restriction of an operation is an operation. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 19-Oct-2021.) |
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦)) & ⊢ (𝜑 → 𝑌 ⊆ 𝑋) & ⊢ (𝜑 → 𝐹:(𝑌 × 𝑌)⟶𝑅) & ⊢ (𝜑 → 𝐺:(𝑋 × 𝑋)⟶𝑆) ⇒ ⊢ (𝜑 → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌))) | ||
Theorem | oprssov 7578 | The value of a member of the domain of a subclass of an operation. (Contributed by NM, 23-Aug-2007.) |
⊢ (((Fun 𝐹 ∧ 𝐺 Fn (𝐶 × 𝐷) ∧ 𝐺 ⊆ 𝐹) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵)) | ||
Theorem | fovcdm 7579 | An operation's value belongs to its codomain. (Contributed by NM, 27-Aug-2006.) |
⊢ ((𝐹:(𝑅 × 𝑆)⟶𝐶 ∧ 𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝐶) | ||
Theorem | fovcdmda 7580 | An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) |
⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶) ⇒ ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) ∈ 𝐶) | ||
Theorem | fovcdmd 7581 | An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) |
⊢ (𝜑 → 𝐹:(𝑅 × 𝑆)⟶𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝑅) & ⊢ (𝜑 → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝐶) | ||
Theorem | fnrnov 7582* | The range of an operation expressed as a collection of the operation's values. (Contributed by NM, 29-Oct-2006.) |
⊢ (𝐹 Fn (𝐴 × 𝐵) → ran 𝐹 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦)}) | ||
Theorem | foov 7583* | An onto mapping of an operation expressed in terms of operation values. (Contributed by NM, 29-Oct-2006.) |
⊢ (𝐹:(𝐴 × 𝐵)–onto→𝐶 ↔ (𝐹:(𝐴 × 𝐵)⟶𝐶 ∧ ∀𝑧 ∈ 𝐶 ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 = (𝑥𝐹𝑦))) | ||
Theorem | fnovrn 7584 | An operation's value belongs to its range. (Contributed by NM, 10-Feb-2007.) |
⊢ ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐶𝐹𝐷) ∈ ran 𝐹) | ||
Theorem | ovelrn 7585* | A member of an operation's range is a value of the operation. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 30-Jan-2014.) |
⊢ (𝐹 Fn (𝐴 × 𝐵) → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝐶 = (𝑥𝐹𝑦))) | ||
Theorem | funimassov 7586* | Membership relation for the values of a function whose image is a subclass. (Contributed by Mario Carneiro, 23-Dec-2013.) |
⊢ ((Fun 𝐹 ∧ (𝐴 × 𝐵) ⊆ dom 𝐹) → ((𝐹 “ (𝐴 × 𝐵)) ⊆ 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝐹𝑦) ∈ 𝐶)) | ||
Theorem | ovelimab 7587* | Operation value in an image. (Contributed by Mario Carneiro, 23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.) |
⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 × 𝐶) ⊆ 𝐴) → (𝐷 ∈ (𝐹 “ (𝐵 × 𝐶)) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐷 = (𝑥𝐹𝑦))) | ||
Theorem | ovima0 7588 | An operation value is a member of the image plus null. (Contributed by Thierry Arnoux, 25-Jun-2019.) |
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑅𝑌) ∈ ((𝑅 “ (𝐴 × 𝐵)) ∪ {∅})) | ||
Theorem | ovconst2 7589 | The value of a constant operation. (Contributed by NM, 5-Nov-2006.) |
⊢ 𝐶 ∈ V ⇒ ⊢ ((𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐵) → (𝑅((𝐴 × 𝐵) × {𝐶})𝑆) = 𝐶) | ||
Theorem | oprssdm 7590* | Domain of closure of an operation. (Contributed by NM, 24-Aug-1995.) |
⊢ ¬ ∅ ∈ 𝑆 & ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) ∈ 𝑆) ⇒ ⊢ (𝑆 × 𝑆) ⊆ dom 𝐹 | ||
Theorem | nssdmovg 7591 | The value of an operation outside its domain. (Contributed by Alexander van der Vekens, 7-Sep-2017.) |
⊢ ((dom 𝐹 ⊆ (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅) | ||
Theorem | ndmovg 7592 | The value of an operation outside its domain. (Contributed by NM, 28-Mar-2008.) |
⊢ ((dom 𝐹 = (𝑅 × 𝑆) ∧ ¬ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴𝐹𝐵) = ∅) | ||
Theorem | ndmov 7593 | The value of an operation outside its domain. (Contributed by NM, 24-Aug-1995.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) ⇒ ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = ∅) | ||
Theorem | ndmovcl 7594 | The closure of an operation outside its domain, when the domain includes the empty set. This technical lemma can make the operation more convenient to work in some cases. It is dependent on our particular definitions of operation value, function value, and ordered pair. (Contributed by NM, 24-Sep-2004.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) & ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) ∈ 𝑆) & ⊢ ∅ ∈ 𝑆 ⇒ ⊢ (𝐴𝐹𝐵) ∈ 𝑆 | ||
Theorem | ndmovrcl 7595 | Reverse closure law, when an operation's domain doesn't contain the empty set. (Contributed by NM, 3-Feb-1996.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) & ⊢ ¬ ∅ ∈ 𝑆 ⇒ ⊢ ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) | ||
Theorem | ndmovcom 7596 | Any operation is commutative outside its domain. (Contributed by NM, 24-Aug-1995.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) ⇒ ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴𝐹𝐵) = (𝐵𝐹𝐴)) | ||
Theorem | ndmovass 7597 | Any operation is associative outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) & ⊢ ¬ ∅ ∈ 𝑆 ⇒ ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) | ||
Theorem | ndmovdistr 7598 | Any operation is distributive outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) & ⊢ ¬ ∅ ∈ 𝑆 & ⊢ dom 𝐺 = (𝑆 × 𝑆) ⇒ ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝐺(𝐵𝐹𝐶)) = ((𝐴𝐺𝐵)𝐹(𝐴𝐺𝐶))) | ||
Theorem | ndmovord 7599 | Elimination of redundant antecedents in an ordering law. (Contributed by NM, 7-Mar-1996.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) & ⊢ 𝑅 ⊆ (𝑆 × 𝑆) & ⊢ ¬ ∅ ∈ 𝑆 & ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) ⇒ ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) | ||
Theorem | ndmovordi 7600 | Elimination of redundant antecedent in an ordering law. (Contributed by NM, 25-Jun-1998.) |
⊢ dom 𝐹 = (𝑆 × 𝑆) & ⊢ 𝑅 ⊆ (𝑆 × 𝑆) & ⊢ ¬ ∅ ∈ 𝑆 & ⊢ (𝐶 ∈ 𝑆 → (𝐴𝑅𝐵 ↔ (𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵))) ⇒ ⊢ ((𝐶𝐹𝐴)𝑅(𝐶𝐹𝐵) → 𝐴𝑅𝐵) |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |