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Type | Label | Description |
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Statement | ||
Theorem | f1cofveqaeqALT 7201 | Alternate proof of f1cofveqaeq 7200, 1 essential step shorter, but having more bytes (305 versus 282). (Contributed by AV, 3-Feb-2021.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌)) → 𝑋 = 𝑌)) | ||
Theorem | 2f1fvneq 7202 | If two one-to-one functions are applied on different arguments, also the values are different. (Contributed by Alexander van der Vekens, 25-Jan-2018.) |
⊢ (((𝐸:𝐷–1-1→𝑅 ∧ 𝐹:𝐶–1-1→𝐷) ∧ (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) ∧ 𝐴 ≠ 𝐵) → (((𝐸‘(𝐹‘𝐴)) = 𝑋 ∧ (𝐸‘(𝐹‘𝐵)) = 𝑌) → 𝑋 ≠ 𝑌)) | ||
Theorem | f1mpt 7203* | Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶) & ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) ⇒ ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝐶 = 𝐷 → 𝑥 = 𝑦))) | ||
Theorem | f1fveq 7204 | Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.) |
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) ↔ 𝐶 = 𝐷)) | ||
Theorem | f1elima 7205 | Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ 𝑋 ∈ 𝑌)) | ||
Theorem | f1imass 7206 | Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ↔ 𝐶 ⊆ 𝐷)) | ||
Theorem | f1imaeq 7207 | Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ 𝐶 = 𝐷)) | ||
Theorem | f1imapss 7208 | Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐴)) → ((𝐹 “ 𝐶) ⊊ (𝐹 “ 𝐷) ↔ 𝐶 ⊊ 𝐷)) | ||
Theorem | fpropnf1 7209 | A function, given by an unordered pair of ordered pairs, which is not injective/one-to-one. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.) |
⊢ 𝐹 = {⟨𝑋, 𝑍⟩, ⟨𝑌, 𝑍⟩} ⇒ ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ 𝑋 ≠ 𝑌) → (Fun 𝐹 ∧ ¬ Fun ◡𝐹)) | ||
Theorem | f1dom3fv3dif 7210 | The function values for a 1-1 function from a set with three different elements are different. (Contributed by AV, 20-Mar-2019.) |
⊢ (𝜑 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) & ⊢ (𝜑 → 𝐹:{𝐴, 𝐵, 𝐶}–1-1→𝑅) ⇒ ⊢ (𝜑 → ((𝐹‘𝐴) ≠ (𝐹‘𝐵) ∧ (𝐹‘𝐴) ≠ (𝐹‘𝐶) ∧ (𝐹‘𝐵) ≠ (𝐹‘𝐶))) | ||
Theorem | f1dom3el3dif 7211* | The codomain of a 1-1 function from a set with three different elements has (at least) three different elements. (Contributed by AV, 20-Mar-2019.) |
⊢ (𝜑 → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍)) & ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶)) & ⊢ (𝜑 → 𝐹:{𝐴, 𝐵, 𝐶}–1-1→𝑅) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑅 ∃𝑧 ∈ 𝑅 (𝑥 ≠ 𝑦 ∧ 𝑥 ≠ 𝑧 ∧ 𝑦 ≠ 𝑧)) | ||
Theorem | dff14a 7212* | A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018.) |
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 → (𝐹‘𝑥) ≠ (𝐹‘𝑦)))) | ||
Theorem | dff14b 7213* | A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018.) |
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹‘𝑥) ≠ (𝐹‘𝑦))) | ||
Theorem | f12dfv 7214 | A one-to-one function with a domain with at least two different elements in terms of function values. (Contributed by Alexander van der Vekens, 2-Mar-2018.) |
⊢ 𝐴 = {𝑋, 𝑌} ⇒ ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉) ∧ 𝑋 ≠ 𝑌) → (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ (𝐹‘𝑋) ≠ (𝐹‘𝑌)))) | ||
Theorem | f13dfv 7215 | A one-to-one function with a domain with at least three different elements in terms of function values. (Contributed by Alexander van der Vekens, 26-Jan-2018.) |
⊢ 𝐴 = {𝑋, 𝑌, 𝑍} ⇒ ⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ((𝐹‘𝑋) ≠ (𝐹‘𝑌) ∧ (𝐹‘𝑋) ≠ (𝐹‘𝑍) ∧ (𝐹‘𝑌) ≠ (𝐹‘𝑍))))) | ||
Theorem | dff1o6 7216* | A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.) |
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))) | ||
Theorem | f1ocnvfv1 7217 | The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → (◡𝐹‘(𝐹‘𝐶)) = 𝐶) | ||
Theorem | f1ocnvfv2 7218 | The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝐶)) = 𝐶) | ||
Theorem | f1ocnvfv 7219 | Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.) |
⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = 𝐶)) | ||
Theorem | f1ocnvfvb 7220 | Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.) |
⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → ((𝐹‘𝐶) = 𝐷 ↔ (◡𝐹‘𝐷) = 𝐶)) | ||
Theorem | nvof1o 7221 | An involution is a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.) |
⊢ ((𝐹 Fn 𝐴 ∧ ◡𝐹 = 𝐹) → 𝐹:𝐴–1-1-onto→𝐴) | ||
Theorem | nvocnv 7222* | The converse of an involution is the function itself. (Contributed by Thierry Arnoux, 7-May-2019.) |
⊢ ((𝐹:𝐴⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘(𝐹‘𝑥)) = 𝑥) → ◡𝐹 = 𝐹) | ||
Theorem | f1cdmsn 7223* | If a one-to-one function with a nonempty domain has a singleton as its codomain, its domain must also be a singleton. (Contributed by BTernaryTau, 1-Dec-2024.) |
⊢ ((𝐹:𝐴–1-1→{𝐵} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥}) | ||
Theorem | fsnex 7224* | Relate a function with a singleton as domain and one variable. (Contributed by Thierry Arnoux, 12-Jul-2020.) |
⊢ (𝑥 = (𝑓‘𝐴) → (𝜓 ↔ 𝜑)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (∃𝑓(𝑓:{𝐴}⟶𝐷 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐷 𝜓)) | ||
Theorem | f1prex 7225* | Relate a one-to-one function with a pair as domain and two different variables. (Contributed by Thierry Arnoux, 12-Jul-2020.) |
⊢ (𝑥 = (𝑓‘𝐴) → (𝜓 ↔ 𝜒)) & ⊢ (𝑦 = (𝑓‘𝐵) → (𝜒 ↔ 𝜑)) ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) → (∃𝑓(𝑓:{𝐴, 𝐵}–1-1→𝐷 ∧ 𝜑) ↔ ∃𝑥 ∈ 𝐷 ∃𝑦 ∈ 𝐷 (𝑥 ≠ 𝑦 ∧ 𝜓))) | ||
Theorem | f1ocnvdm 7226 | The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.) |
⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐵) → (◡𝐹‘𝐶) ∈ 𝐴) | ||
Theorem | f1ocnvfvrneq 7227 | If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.) |
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝐶 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹)) → ((◡𝐹‘𝐶) = (◡𝐹‘𝐷) → 𝐶 = 𝐷)) | ||
Theorem | fcof1 7228 | An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝑅 ∘ 𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴–1-1→𝐵) | ||
Theorem | fcofo 7229 | An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑆:𝐵⟶𝐴 ∧ (𝐹 ∘ 𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴–onto→𝐵) | ||
Theorem | cbvfo 7230* | Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
⊢ ((𝐹‘𝑥) = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐹:𝐴–onto→𝐵 → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓)) | ||
Theorem | cbvexfo 7231* | Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) |
⊢ ((𝐹‘𝑥) = 𝑦 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐹:𝐴–onto→𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓)) | ||
Theorem | cocan1 7232 | An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.) |
⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐻:𝐴⟶𝐵 ∧ 𝐾:𝐴⟶𝐵) → ((𝐹 ∘ 𝐻) = (𝐹 ∘ 𝐾) ↔ 𝐻 = 𝐾)) | ||
Theorem | cocan2 7233 | A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐻 Fn 𝐵 ∧ 𝐾 Fn 𝐵) → ((𝐻 ∘ 𝐹) = (𝐾 ∘ 𝐹) ↔ 𝐻 = 𝐾)) | ||
Theorem | fcof1oinvd 7234 | Show that a function is the inverse of a bijective function if their composition is the identity function. Formerly part of proof of fcof1o 7237. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.) |
⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) & ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵)) ⇒ ⊢ (𝜑 → ◡𝐹 = 𝐺) | ||
Theorem | fcof1od 7235 | A function is bijective if a "retraction" and a "section" exist, see comments for fcof1 7228 and fcofo 7229. Formerly part of proof of fcof1o 7237. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) & ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)) & ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵)) ⇒ ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | ||
Theorem | 2fcoidinvd 7236 | Show that a function is the inverse of a function if their compositions are the identity functions. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) & ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)) & ⊢ (𝜑 → (𝐹 ∘ 𝐺) = ( I ↾ 𝐵)) ⇒ ⊢ (𝜑 → ◡𝐹 = 𝐺) | ||
Theorem | fcof1o 7237 | Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.) (Proof shortened by AV, 15-Dec-2019.) |
⊢ (((𝐹:𝐴⟶𝐵 ∧ 𝐺:𝐵⟶𝐴) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ 𝐵) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = 𝐺)) | ||
Theorem | 2fvcoidd 7238* | Show that the composition of two functions is the identity function by applying both functions to each value of the domain of the first function. (Contributed by AV, 15-Dec-2019.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) & ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 (𝐺‘(𝐹‘𝑎)) = 𝑎) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)) | ||
Theorem | 2fvidf1od 7239* | A function is bijective if it has an inverse function. (Contributed by AV, 15-Dec-2019.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) & ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 (𝐺‘(𝐹‘𝑎)) = 𝑎) & ⊢ (𝜑 → ∀𝑏 ∈ 𝐵 (𝐹‘(𝐺‘𝑏)) = 𝑏) ⇒ ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | ||
Theorem | 2fvidinvd 7240* | Show that two functions are inverse to each other by applying them twice to each value of their domains. (Contributed by AV, 13-Dec-2019.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) & ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 (𝐺‘(𝐹‘𝑎)) = 𝑎) & ⊢ (𝜑 → ∀𝑏 ∈ 𝐵 (𝐹‘(𝐺‘𝑏)) = 𝑏) ⇒ ⊢ (𝜑 → ◡𝐹 = 𝐺) | ||
Theorem | foeqcnvco 7241 | Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.) |
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 = 𝐺 ↔ (𝐹 ∘ ◡𝐺) = ( I ↾ 𝐵))) | ||
Theorem | f1eqcocnv 7242 | Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.) (Proof shortened by Wolf Lammen, 29-May-2024.) |
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴))) | ||
Theorem | f1eqcocnvOLD 7243 | Obsolete version of f1eqcocnv 7242 as of 29-May-2024. (Contributed by Stefan O'Rear, 12-Feb-2015.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 = 𝐺 ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ 𝐴))) | ||
Theorem | fveqf1o 7244 | Given a bijection 𝐹, produce another bijection 𝐺 which additionally maps two specified points. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ 𝐺 = (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝐶, (◡𝐹‘𝐷)})) ∪ {⟨𝐶, (◡𝐹‘𝐷)⟩, ⟨(◡𝐹‘𝐷), 𝐶⟩})) ⇒ ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) → (𝐺:𝐴–1-1-onto→𝐵 ∧ (𝐺‘𝐶) = 𝐷)) | ||
Theorem | nf1const 7245 | A constant function from at least two elements is not one-to-one. (Contributed by AV, 30-Mar-2024.) |
⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ 𝐹:𝐴–1-1→𝐶) | ||
Theorem | nf1oconst 7246 | A constant function from at least two elements is not bijective. (Contributed by AV, 30-Mar-2024.) |
⊢ ((𝐹:𝐴⟶{𝐵} ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → ¬ 𝐹:𝐴–1-1-onto→𝐶) | ||
Theorem | f1ofvswap 7247 | Swapping two values in a bijection between two classes yields another bijection between those classes. (Contributed by BTernaryTau, 31-Aug-2024.) |
⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩}):𝐴–1-1-onto→𝐵) | ||
Theorem | fliftrel 7248* | 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → 𝐹 ⊆ (𝑅 × 𝑆)) | ||
Theorem | fliftel 7249* | Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶 = 𝐴 ∧ 𝐷 = 𝐵))) | ||
Theorem | fliftel1 7250* | Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴𝐹𝐵) | ||
Theorem | fliftcnv 7251* | Converse of the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → ◡𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐵, 𝐴⟩)) | ||
Theorem | fliftfun 7252* | The function 𝐹 is the unique function defined by 𝐹‘𝐴 = 𝐵, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐶) & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐴 = 𝐶 → 𝐵 = 𝐷))) | ||
Theorem | fliftfund 7253* | The function 𝐹 is the unique function defined by 𝐹‘𝐴 = 𝐵, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) & ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐶) & ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐷) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ 𝐴 = 𝐶)) → 𝐵 = 𝐷) ⇒ ⊢ (𝜑 → Fun 𝐹) | ||
Theorem | fliftfuns 7254* | The function 𝐹 is the unique function defined by 𝐹‘𝐴 = 𝐵, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → (Fun 𝐹 ↔ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑧 / 𝑥⦌𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵))) | ||
Theorem | fliftf 7255* | The domain and range of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) ⇒ ⊢ (𝜑 → (Fun 𝐹 ↔ 𝐹:ran (𝑥 ∈ 𝑋 ↦ 𝐴)⟶𝑆)) | ||
Theorem | fliftval 7256* | The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ ⟨𝐴, 𝐵⟩) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑅) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑆) & ⊢ (𝑥 = 𝑌 → 𝐴 = 𝐶) & ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐷) & ⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ ((𝜑 ∧ 𝑌 ∈ 𝑋) → (𝐹‘𝐶) = 𝐷) | ||
Theorem | isoeq1 7257 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
⊢ (𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) | ||
Theorem | isoeq2 7258 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
⊢ (𝑅 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑇, 𝑆 (𝐴, 𝐵))) | ||
Theorem | isoeq3 7259 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
⊢ (𝑆 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑇 (𝐴, 𝐵))) | ||
Theorem | isoeq4 7260 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
⊢ (𝐴 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵))) | ||
Theorem | isoeq5 7261 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
⊢ (𝐵 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐶))) | ||
Theorem | nfiso 7262 | Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ Ⅎ𝑥𝐻 & ⊢ Ⅎ𝑥𝑅 & ⊢ Ⅎ𝑥𝑆 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) | ||
Theorem | isof1o 7263 | An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵) | ||
Theorem | isof1oidb 7264 | A function is a bijection iff it is an isomorphism regarding the identity relation. (Contributed by AV, 9-May-2021.) |
⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom I , I (𝐴, 𝐵)) | ||
Theorem | isof1oopb 7265 | A function is a bijection iff it is an isomorphism regarding the universal class of ordered pairs as relations. (Contributed by AV, 9-May-2021.) |
⊢ (𝐻:𝐴–1-1-onto→𝐵 ↔ 𝐻 Isom (V × V), (V × V)(𝐴, 𝐵)) | ||
Theorem | isorel 7266 | An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.) |
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷))) | ||
Theorem | soisores 7267* | Express the condition of isomorphism on two strict orders for a function's restriction. (Contributed by Mario Carneiro, 22-Jan-2015.) |
⊢ (((𝑅 Or 𝐵 ∧ 𝑆 Or 𝐶) ∧ (𝐹:𝐵⟶𝐶 ∧ 𝐴 ⊆ 𝐵)) → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹 “ 𝐴)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐹‘𝑥)𝑆(𝐹‘𝑦)))) | ||
Theorem | soisoi 7268* | Infer isomorphism from one direction of an order proof for isomorphisms between strict orders. (Contributed by Stefan O'Rear, 2-Nov-2014.) |
⊢ (((𝑅 Or 𝐴 ∧ 𝑆 Po 𝐵) ∧ (𝐻:𝐴–onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)) | ||
Theorem | isoid 7269 | Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
⊢ ( I ↾ 𝐴) Isom 𝑅, 𝑅 (𝐴, 𝐴) | ||
Theorem | isocnv 7270 | Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ◡𝐻 Isom 𝑆, 𝑅 (𝐵, 𝐴)) | ||
Theorem | isocnv2 7271 | Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵)) | ||
Theorem | isocnv3 7272 | Complementation law for isomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.) |
⊢ 𝐶 = ((𝐴 × 𝐴) ∖ 𝑅) & ⊢ 𝐷 = ((𝐵 × 𝐵) ∖ 𝑆) ⇒ ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝐶, 𝐷 (𝐴, 𝐵)) | ||
Theorem | isores2 7273 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, (𝑆 ∩ (𝐵 × 𝐵))(𝐴, 𝐵)) | ||
Theorem | isores1 7274 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom (𝑅 ∩ (𝐴 × 𝐴)), 𝑆(𝐴, 𝐵)) | ||
Theorem | isores3 7275 | Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾 ⊆ 𝐴 ∧ 𝑋 = (𝐻 “ 𝐾)) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋)) | ||
Theorem | isotr 7276 | Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑆, 𝑇 (𝐵, 𝐶)) → (𝐺 ∘ 𝐻) Isom 𝑅, 𝑇 (𝐴, 𝐶)) | ||
Theorem | isomin 7277 | Isomorphisms preserve minimal elements. Note that (◡𝑅 “ {𝐷}) is Takeuti and Zaring's idiom for the initial segment {𝑥 ∣ 𝑥𝑅𝐷}. Proposition 6.31(1) of [TakeutiZaring] p. 33. (Contributed by NM, 19-Apr-2004.) |
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅ ↔ ((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅)) | ||
Theorem | isoini 7278 | Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.) |
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝐷}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝐷)}))) | ||
Theorem | isoini2 7279 | Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.) |
⊢ 𝐶 = (𝐴 ∩ (◡𝑅 “ {𝑋})) & ⊢ 𝐷 = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑋)})) ⇒ ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷)) | ||
Theorem | isofrlem 7280* | Lemma for isofr 7282. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.) |
⊢ (𝜑 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)) & ⊢ (𝜑 → (𝐻 “ 𝑥) ∈ V) ⇒ ⊢ (𝜑 → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴)) | ||
Theorem | isoselem 7281* | Lemma for isose 7283. (Contributed by Mario Carneiro, 23-Jun-2015.) |
⊢ (𝜑 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)) & ⊢ (𝜑 → (𝐻 “ 𝑥) ∈ V) ⇒ ⊢ (𝜑 → (𝑅 Se 𝐴 → 𝑆 Se 𝐵)) | ||
Theorem | isofr 7282 | An isomorphism preserves well-foundedness. Proposition 6.32(1) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵)) | ||
Theorem | isose 7283 | An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐵)) | ||
Theorem | isofr2 7284 | A weak form of isofr 7282 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.) |
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵 ∈ 𝑉) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴)) | ||
Theorem | isopolem 7285 | Lemma for isopo 7286. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵 → 𝑅 Po 𝐴)) | ||
Theorem | isopo 7286 | An isomorphism preserves the property of being a partial order. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Po 𝐴 ↔ 𝑆 Po 𝐵)) | ||
Theorem | isosolem 7287 | Lemma for isoso 7288. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵 → 𝑅 Or 𝐴)) | ||
Theorem | isoso 7288 | An isomorphism preserves the property of being a strict total order. (Contributed by Stefan O'Rear, 16-Nov-2014.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵)) | ||
Theorem | isowe 7289 | An isomorphism preserves the property of being a well-ordering. Proposition 6.32(3) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵)) | ||
Theorem | isowe2 7290* | A weak form of isowe 7289 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.) |
⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻 “ 𝑥) ∈ V) → (𝑆 We 𝐵 → 𝑅 We 𝐴)) | ||
Theorem | f1oiso 7291* | Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. Proposition 6.33 of [TakeutiZaring] p. 34. (Contributed by NM, 30-Apr-2004.) |
⊢ ((𝐻:𝐴–1-1-onto→𝐵 ∧ 𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝑧 = (𝐻‘𝑥) ∧ 𝑤 = (𝐻‘𝑦)) ∧ 𝑥𝑅𝑦)}) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)) | ||
Theorem | f1oiso2 7292* | Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. (Contributed by Mario Carneiro, 9-Mar-2013.) |
⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (◡𝐻‘𝑥)𝑅(◡𝐻‘𝑦))} ⇒ ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)) | ||
Theorem | f1owe 7293* | Well-ordering of isomorphic relations. (Contributed by NM, 4-Mar-1997.) |
⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹‘𝑥)𝑆(𝐹‘𝑦)} ⇒ ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝑆 We 𝐵 → 𝑅 We 𝐴)) | ||
Theorem | weniso 7294 | A set-like well-ordering has no nontrivial automorphisms. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Revised by Mario Carneiro, 25-Jun-2015.) |
⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ∧ 𝐹 Isom 𝑅, 𝑅 (𝐴, 𝐴)) → 𝐹 = ( I ↾ 𝐴)) | ||
Theorem | weisoeq 7295 | Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso 7897. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺) | ||
Theorem | weisoeq2 7296 | Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso2 7898. (Contributed by Mario Carneiro, 25-Jun-2015.) |
⊢ (((𝑆 We 𝐵 ∧ 𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺) | ||
Theorem | knatar 7297* | The Knaster-Tarski theorem says that every monotone function over a complete lattice has a (least) fixpoint. Here we specialize this theorem to the case when the lattice is the powerset lattice 𝒫 𝐴. (Contributed by Mario Carneiro, 11-Jun-2015.) |
⊢ 𝑋 = ∩ {𝑧 ∈ 𝒫 𝐴 ∣ (𝐹‘𝑧) ⊆ 𝑧} ⇒ ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐹‘𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴∀𝑦 ∈ 𝒫 𝑥(𝐹‘𝑦) ⊆ (𝐹‘𝑥)) → (𝑋 ⊆ 𝐴 ∧ (𝐹‘𝑋) = 𝑋)) | ||
Theorem | fvresval 7298 | The value of a restricted function at a class is either the empty set or the value of the unrestricted function at that class. (Contributed by Scott Fenton, 4-Sep-2011.) |
⊢ (((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴) ∨ ((𝐹 ↾ 𝐵)‘𝐴) = ∅) | ||
Theorem | funeldmb 7299 | If ∅ is not part of the range of a function 𝐹, then 𝐴 is in the domain of 𝐹 iff (𝐹‘𝐴) ≠ ∅. (Contributed by Scott Fenton, 7-Dec-2021.) |
⊢ ((Fun 𝐹 ∧ ¬ ∅ ∈ ran 𝐹) → (𝐴 ∈ dom 𝐹 ↔ (𝐹‘𝐴) ≠ ∅)) | ||
Theorem | eqfunresadj 7300 | Law for adjoining an element to restrictions of functions. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → (𝐹 ↾ (𝑋 ∪ {𝑌})) = (𝐺 ↾ (𝑋 ∪ {𝑌}))) |
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