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Theorem List for Metamath Proof Explorer - 6501-6600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremf1cofveqaeqALT 6501 Alternate proof of f1cofveqaeq 6500, 1 essential step shorter, but having more bytes (305 vs. 282). (Contributed by AV, 3-Feb-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
(((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → 𝑋 = 𝑌))

Theorem2f1fvneq 6502 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:𝐷1-1𝑅𝐹:𝐶1-1𝐷) ∧ (𝐴𝐶𝐵𝐶) ∧ 𝐴𝐵) → (((𝐸‘(𝐹𝐴)) = 𝑋 ∧ (𝐸‘(𝐹𝐵)) = 𝑌) → 𝑋𝑌))

Theoremf1mpt 6503* Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
𝐹 = (𝑥𝐴𝐶)    &   (𝑥 = 𝑦𝐶 = 𝐷)       (𝐹:𝐴1-1𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝐶 = 𝐷𝑥 = 𝑦)))

Theoremf1fveq 6504 Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.)
((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) ↔ 𝐶 = 𝐷))

Theoremf1elima 6505 Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ 𝑋𝑌))

Theoremf1imass 6506 Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)
((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) ⊆ (𝐹𝐷) ↔ 𝐶𝐷))

Theoremf1imaeq 6507 Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)
((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) ↔ 𝐶 = 𝐷))

Theoremf1imapss 6508 Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.)
((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) ⊊ (𝐹𝐷) ↔ 𝐶𝐷))

Theoremfpropnf1 6509 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⟩}       (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ 𝑋𝑌) → (Fun 𝐹 ∧ ¬ Fun 𝐹))

Theoremf1dom3fv3dif 6510 The function values for a 1-1 function from a set with three different elements are different. (Contributed by AV, 20-Mar-2019.)
(𝜑 → (𝐴𝑋𝐵𝑌𝐶𝑍))    &   (𝜑 → (𝐴𝐵𝐴𝐶𝐵𝐶))    &   (𝜑𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅)       (𝜑 → ((𝐹𝐴) ≠ (𝐹𝐵) ∧ (𝐹𝐴) ≠ (𝐹𝐶) ∧ (𝐹𝐵) ≠ (𝐹𝐶)))

Theoremf1dom3el3dif 6511* The range 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 → (𝐴𝑋𝐵𝑌𝐶𝑍))    &   (𝜑 → (𝐴𝐵𝐴𝐶𝐵𝐶))    &   (𝜑𝐹:{𝐴, 𝐵, 𝐶}–1-1𝑅)       (𝜑 → ∃𝑥𝑅𝑦𝑅𝑧𝑅 (𝑥𝑦𝑥𝑧𝑦𝑧))

Theoremdff14a 6512* A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
(𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦 → (𝐹𝑥) ≠ (𝐹𝑦))))

Theoremdff14b 6513* A one-to-one function in terms of different function values for different arguments. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
(𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ (𝐴 ∖ {𝑥})(𝐹𝑥) ≠ (𝐹𝑦)))

Theoremf12dfv 6514 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}       (((𝑋𝑈𝑌𝑉) ∧ 𝑋𝑌) → (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ (𝐹𝑋) ≠ (𝐹𝑌))))

Theoremf13dfv 6515 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}       (((𝑋𝑈𝑌𝑉𝑍𝑊) ∧ (𝑋𝑌𝑋𝑍𝑌𝑍)) → (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ((𝐹𝑋) ≠ (𝐹𝑌) ∧ (𝐹𝑋) ≠ (𝐹𝑍) ∧ (𝐹𝑌) ≠ (𝐹𝑍)))))

Theoremdff1o6 6516* A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.)
(𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))

Theoremf1ocnvfv1 6517 The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
((𝐹:𝐴1-1-onto𝐵𝐶𝐴) → (𝐹‘(𝐹𝐶)) = 𝐶)

Theoremf1ocnvfv2 6518 The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.)
((𝐹:𝐴1-1-onto𝐵𝐶𝐵) → (𝐹‘(𝐹𝐶)) = 𝐶)

Theoremf1ocnvfv 6519 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:𝐴1-1-onto𝐵𝐶𝐴) → ((𝐹𝐶) = 𝐷 → (𝐹𝐷) = 𝐶))

Theoremf1ocnvfvb 6520 Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.)
((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → ((𝐹𝐶) = 𝐷 ↔ (𝐹𝐷) = 𝐶))

Theoremnvof1o 6521 An involution is a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1-onto𝐴)

Theoremnvocnv 6522* The converse of an involution is the function itself. (Contributed by Thierry Arnoux, 7-May-2019.)
((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = 𝐹)

Theoremfsnex 6523* Relate a function with a singleton as domain and one variable. (Contributed by Thierry Arnoux, 12-Jul-2020.)
(𝑥 = (𝑓𝐴) → (𝜓𝜑))       (𝐴𝑉 → (∃𝑓(𝑓:{𝐴}⟶𝐷𝜑) ↔ ∃𝑥𝐷 𝜓))

Theoremf1prex 6524* Relate a one-to-one function with a pair as domain and two different variables. (Contributed by Thierry Arnoux, 12-Jul-2020.)
(𝑥 = (𝑓𝐴) → (𝜓𝜒))    &   (𝑦 = (𝑓𝐵) → (𝜒𝜑))       ((𝐴𝑉𝐵𝑊𝐴𝐵) → (∃𝑓(𝑓:{𝐴, 𝐵}–1-1𝐷𝜑) ↔ ∃𝑥𝐷𝑦𝐷 (𝑥𝑦𝜓)))

Theoremf1ocnvdm 6525 The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.)
((𝐹:𝐴1-1-onto𝐵𝐶𝐵) → (𝐹𝐶) ∈ 𝐴)

Theoremf1ocnvfvrneq 6526 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:𝐴1-1𝐵 ∧ (𝐶 ∈ ran 𝐹𝐷 ∈ ran 𝐹)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))

Theoremfcof1 6527 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:𝐴𝐵 ∧ (𝑅𝐹) = ( I ↾ 𝐴)) → 𝐹:𝐴1-1𝐵)

Theoremfcofo 6528 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:𝐴𝐵𝑆:𝐵𝐴 ∧ (𝐹𝑆) = ( I ↾ 𝐵)) → 𝐹:𝐴onto𝐵)

Theoremcbvfo 6529* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
((𝐹𝑥) = 𝑦 → (𝜑𝜓))       (𝐹:𝐴onto𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐵 𝜓))

Theoremcbvexfo 6530* Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.)
((𝐹𝑥) = 𝑦 → (𝜑𝜓))       (𝐹:𝐴onto𝐵 → (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐵 𝜓))

Theoremcocan1 6531 An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.)
((𝐹:𝐵1-1𝐶𝐻:𝐴𝐵𝐾:𝐴𝐵) → ((𝐹𝐻) = (𝐹𝐾) ↔ 𝐻 = 𝐾))

Theoremcocan2 6532 A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
((𝐹:𝐴onto𝐵𝐻 Fn 𝐵𝐾 Fn 𝐵) → ((𝐻𝐹) = (𝐾𝐹) ↔ 𝐻 = 𝐾))

Theoremfcof1oinvd 6533 Show that a function is the inverse of a bijective function if their composition is the identity function. Formerly part of proof of fcof1o 6536. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.)
(𝜑𝐹:𝐴1-1-onto𝐵)    &   (𝜑𝐺:𝐵𝐴)    &   (𝜑 → (𝐹𝐺) = ( I ↾ 𝐵))       (𝜑𝐹 = 𝐺)

Theoremfcof1od 6534 A function is bijective if a "retraction" and a "section" exist, see comments for fcof1 6527 and fcofo 6528. Formerly part of proof of fcof1o 6536. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by AV, 15-Dec-2019.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐵𝐴)    &   (𝜑 → (𝐺𝐹) = ( I ↾ 𝐴))    &   (𝜑 → (𝐹𝐺) = ( I ↾ 𝐵))       (𝜑𝐹:𝐴1-1-onto𝐵)

Theorem2fcoidinvd 6535 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.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐵𝐴)    &   (𝜑 → (𝐺𝐹) = ( I ↾ 𝐴))    &   (𝜑 → (𝐹𝐺) = ( I ↾ 𝐵))       (𝜑𝐹 = 𝐺)

Theoremfcof1o 6536 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:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹:𝐴1-1-onto𝐵𝐹 = 𝐺))

Theorem2fvcoidd 6537* 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.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐵𝐴)    &   (𝜑 → ∀𝑎𝐴 (𝐺‘(𝐹𝑎)) = 𝑎)       (𝜑 → (𝐺𝐹) = ( I ↾ 𝐴))

Theorem2fvidf1od 6538* A function is bijective if it has an inverse function. (Contributed by AV, 15-Dec-2019.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐵𝐴)    &   (𝜑 → ∀𝑎𝐴 (𝐺‘(𝐹𝑎)) = 𝑎)    &   (𝜑 → ∀𝑏𝐵 (𝐹‘(𝐺𝑏)) = 𝑏)       (𝜑𝐹:𝐴1-1-onto𝐵)

Theorem2fvidinvd 6539* 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.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐵𝐴)    &   (𝜑 → ∀𝑎𝐴 (𝐺‘(𝐹𝑎)) = 𝑎)    &   (𝜑 → ∀𝑏𝐵 (𝐹‘(𝐺𝑏)) = 𝑏)       (𝜑𝐹 = 𝐺)

Theoremfoeqcnvco 6540 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:𝐴onto𝐵𝐺:𝐴onto𝐵) → (𝐹 = 𝐺 ↔ (𝐹𝐺) = ( I ↾ 𝐵)))

Theoremf1eqcocnv 6541 Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → (𝐹 = 𝐺 ↔ (𝐹𝐺) = ( I ↾ 𝐴)))

Theoremfveqf1o 6542 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, (𝐹𝐷)})) ∪ {⟨𝐶, (𝐹𝐷)⟩, ⟨(𝐹𝐷), 𝐶⟩}))       ((𝐹:𝐴1-1-onto𝐵𝐶𝐴𝐷𝐵) → (𝐺:𝐴1-1-onto𝐵 ∧ (𝐺𝐶) = 𝐷))

Theoremfliftrel 6543* 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑅)    &   ((𝜑𝑥𝑋) → 𝐵𝑆)       (𝜑𝐹 ⊆ (𝑅 × 𝑆))

Theoremfliftel 6544* Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑅)    &   ((𝜑𝑥𝑋) → 𝐵𝑆)       (𝜑 → (𝐶𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶 = 𝐴𝐷 = 𝐵)))

Theoremfliftel1 6545* Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑅)    &   ((𝜑𝑥𝑋) → 𝐵𝑆)       ((𝜑𝑥𝑋) → 𝐴𝐹𝐵)

Theoremfliftcnv 6546* Converse of the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑅)    &   ((𝜑𝑥𝑋) → 𝐵𝑆)       (𝜑𝐹 = ran (𝑥𝑋 ↦ ⟨𝐵, 𝐴⟩))

Theoremfliftfun 6547* The function 𝐹 is the unique function defined by 𝐹𝐴 = 𝐵, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑅)    &   ((𝜑𝑥𝑋) → 𝐵𝑆)    &   (𝑥 = 𝑦𝐴 = 𝐶)    &   (𝑥 = 𝑦𝐵 = 𝐷)       (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑋𝑦𝑋 (𝐴 = 𝐶𝐵 = 𝐷)))

Theoremfliftfund 6548* The function 𝐹 is the unique function defined by 𝐹𝐴 = 𝐵, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑅)    &   ((𝜑𝑥𝑋) → 𝐵𝑆)    &   (𝑥 = 𝑦𝐴 = 𝐶)    &   (𝑥 = 𝑦𝐵 = 𝐷)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝐴 = 𝐶)) → 𝐵 = 𝐷)       (𝜑 → Fun 𝐹)

Theoremfliftfuns 6549* The function 𝐹 is the unique function defined by 𝐹𝐴 = 𝐵, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑅)    &   ((𝜑𝑥𝑋) → 𝐵𝑆)       (𝜑 → (Fun 𝐹 ↔ ∀𝑦𝑋𝑧𝑋 (𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴𝑦 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)))

Theoremfliftf 6550* The domain and range of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑅)    &   ((𝜑𝑥𝑋) → 𝐵𝑆)       (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋𝐴)⟶𝑆))

Theoremfliftval 6551* The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑅)    &   ((𝜑𝑥𝑋) → 𝐵𝑆)    &   (𝑥 = 𝑌𝐴 = 𝐶)    &   (𝑥 = 𝑌𝐵 = 𝐷)    &   (𝜑 → Fun 𝐹)       ((𝜑𝑌𝑋) → (𝐹𝐶) = 𝐷)

Theoremisoeq1 6552 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
(𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵)))

Theoremisoeq2 6553 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
(𝑅 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑇, 𝑆 (𝐴, 𝐵)))

Theoremisoeq3 6554 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
(𝑆 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑇 (𝐴, 𝐵)))

Theoremisoeq4 6555 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
(𝐴 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵)))

Theoremisoeq5 6556 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
(𝐵 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐶)))

Theoremnfiso 6557 Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
𝑥𝐻    &   𝑥𝑅    &   𝑥𝑆    &   𝑥𝐴    &   𝑥𝐵       𝑥 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵)

Theoremisof1o 6558 An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴1-1-onto𝐵)

Theoremisof1oidb 6559 A function is a bijection iff it is an isomorphism regarding the identity relation. (Contributed by AV, 9-May-2021.)
(𝐻:𝐴1-1-onto𝐵𝐻 Isom I , I (𝐴, 𝐵))

Theoremisof1oopb 6560 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:𝐴1-1-onto𝐵𝐻 Isom (V × V), (V × V)(𝐴, 𝐵))

Theoremisorel 6561 An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)
((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝑅𝐷 ↔ (𝐻𝐶)𝑆(𝐻𝐷)))

Theoremsoisores 6562* Express the condition of isomorphism on two strict orders for a function's restriction. (Contributed by Mario Carneiro, 22-Jan-2015.)
(((𝑅 Or 𝐵𝑆 Or 𝐶) ∧ (𝐹:𝐵𝐶𝐴𝐵)) → ((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦))))

Theoremsoisoi 6563* Infer isomorphism from one direction of an order proof for isomorphisms between strict orders. (Contributed by Stefan O'Rear, 2-Nov-2014.)
(((𝑅 Or 𝐴𝑆 Po 𝐵) ∧ (𝐻:𝐴onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐻𝑥)𝑆(𝐻𝑦)))) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))

Theoremisoid 6564 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)

Theoremisocnv 6565 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))

Theoremisocnv2 6566 Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆(𝐴, 𝐵))

Theoremisocnv3 6567 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))

Theoremisores2 6568 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))

Theoremisores1 6569 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))

Theoremisores3 6570 Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)
((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾𝐴𝑋 = (𝐻𝐾)) → (𝐻𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋))

Theoremisotr 6571 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)) → (𝐺𝐻) Isom 𝑅, 𝑇 (𝐴, 𝐶))

Theoremisomin 6572 Isomorphisms preserve minimal elements. Note that (𝑅 “ {𝐷}) is Takeuti and Zaring's idiom for the initial segment {𝑥𝑥𝑅𝐷}. Proposition 6.31(1) of [TakeutiZaring] p. 33. (Contributed by NM, 19-Apr-2004.)
((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → ((𝐶 ∩ (𝑅 “ {𝐷})) = ∅ ↔ ((𝐻𝐶) ∩ (𝑆 “ {(𝐻𝐷)})) = ∅))

Theoremisoini 6573 Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.)
((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐷𝐴) → (𝐻 “ (𝐴 ∩ (𝑅 “ {𝐷}))) = (𝐵 ∩ (𝑆 “ {(𝐻𝐷)})))

Theoremisoini2 6574 Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)
𝐶 = (𝐴 ∩ (𝑅 “ {𝑋}))    &   𝐷 = (𝐵 ∩ (𝑆 “ {(𝐻𝑋)}))       ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋𝐴) → (𝐻𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷))

Theoremisofrlem 6575* Lemma for isofr 6577. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)
(𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑 → (𝐻𝑥) ∈ V)       (𝜑 → (𝑆 Fr 𝐵𝑅 Fr 𝐴))

Theoremisoselem 6576* Lemma for isose 6578. (Contributed by Mario Carneiro, 23-Jun-2015.)
(𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑 → (𝐻𝑥) ∈ V)       (𝜑 → (𝑅 Se 𝐴𝑆 Se 𝐵))

Theoremisofr 6577 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 𝐴𝑆 Fr 𝐵))

Theoremisose 6578 An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴𝑆 Se 𝐵))

Theoremisofr2 6579 A weak form of isofr 6577 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)
((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐵𝑉) → (𝑆 Fr 𝐵𝑅 Fr 𝐴))

Theoremisopolem 6580 Lemma for isopo 6581. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵𝑅 Po 𝐴))

Theoremisopo 6581 An isomorphism preserves partial ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Po 𝐴𝑆 Po 𝐵))

Theoremisosolem 6582 Lemma for isoso 6583. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵𝑅 Or 𝐴))

Theoremisoso 6583 An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴𝑆 Or 𝐵))

Theoremisowe 6584 An isomorphism preserves 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 𝐴𝑆 We 𝐵))

Theoremisowe2 6585* A weak form of isowe 6584 that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014.)
((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ ∀𝑥(𝐻𝑥) ∈ V) → (𝑆 We 𝐵𝑅 We 𝐴))

Theoremf1oiso 6586* 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:𝐴1-1-onto𝐵𝑆 = {⟨𝑧, 𝑤⟩ ∣ ∃𝑥𝐴𝑦𝐴 ((𝑧 = (𝐻𝑥) ∧ 𝑤 = (𝐻𝑦)) ∧ 𝑥𝑅𝑦)}) → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))

Theoremf1oiso2 6587* Any one-to-one onto function determines an isomorphism with an induced relation 𝑆. (Contributed by Mario Carneiro, 9-Mar-2013.)
𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ (𝐻𝑥)𝑅(𝐻𝑦))}       (𝐻:𝐴1-1-onto𝐵𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))

Theoremf1owe 6588* Well-ordering of isomorphic relations. (Contributed by NM, 4-Mar-1997.)
𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝐹𝑥)𝑆(𝐹𝑦)}       (𝐹:𝐴1-1-onto𝐵 → (𝑆 We 𝐵𝑅 We 𝐴))

Theoremweniso 6589 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 𝐴𝑅 Se 𝐴𝐹 Isom 𝑅, 𝑅 (𝐴, 𝐴)) → 𝐹 = ( I ↾ 𝐴))

Theoremweisoeq 6590 Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso 7138. (Contributed by Mario Carneiro, 25-Jun-2015.)
(((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)

Theoremweisoeq2 6591 Thus, there is at most one isomorphism between any two set-like well-ordered classes. Class version of wemoiso2 7139. (Contributed by Mario Carneiro, 25-Jun-2015.)
(((𝑆 We 𝐵𝑆 Se 𝐵) ∧ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐺 Isom 𝑅, 𝑆 (𝐴, 𝐵))) → 𝐹 = 𝐺)

Theoremknatar 6592* 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 ∣ (𝐹𝑧) ⊆ 𝑧}       ((𝐴𝑉 ∧ (𝐹𝐴) ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝑥(𝐹𝑦) ⊆ (𝐹𝑥)) → (𝑋𝐴 ∧ (𝐹𝑋) = 𝑋))

2.3.16  Cantor's Theorem

Theoremcanth 6593 No set 𝐴 is equinumerous to its power set (Cantor's theorem), i.e. no function can map 𝐴 it onto its power set. Compare Theorem 6B(b) of [Enderton] p. 132. For the equinumerosity version, see canth2 8098. Note that 𝐴 must be a set: this theorem does not hold when 𝐴 is too large to be a set; see ncanth 6594 for a counterexample. (Use nex 1729 if you want the form ¬ ∃𝑓𝑓:𝐴onto→𝒫 𝐴.) (Contributed by NM, 7-Aug-1994.) (Proof shortened by Mario Carneiro, 7-Jun-2016.)
𝐴 ∈ V        ¬ 𝐹:𝐴onto→𝒫 𝐴

Theoremncanth 6594 Cantor's theorem fails for the universal class (which is not a set but a proper class by vprc 4787). Specifically, the identity function maps the universe onto its power class. Compare canth 6593 that works for sets. See also the remark in ru 3428 about NF, in which Cantor's theorem fails for sets that are "too large." This theorem gives some intuition behind that failure: in NF the universal class is a set, and it equals its own power set. (Contributed by NM, 29-Jun-2004.)
I :V–onto→𝒫 V

2.3.17  Restricted iota (description binder)

Syntaxcrio 6595 Extend class notation with restricted description binder.
class (𝑥𝐴 𝜑)

Definitiondf-riota 6596 Define restricted description binder. In case there is no unique 𝑥 such that (𝑥𝐴𝜑) holds, it evaluates to the empty set. See also comments for df-iota 5839. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.)
(𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))

Theoremriotaeqdv 6597* Formula-building deduction rule for iota. (Contributed by NM, 15-Sep-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜓))

Theoremriotabidv 6598* Formula-building deduction rule for restricted iota. (Contributed by NM, 15-Sep-2011.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))

Theoremriotaeqbidv 6599* Equality deduction for restricted universal quantifier. (Contributed by NM, 15-Sep-2011.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜒))

Theoremriotaex 6600 Restricted iota is a set. (Contributed by NM, 15-Sep-2011.)
(𝑥𝐴 𝜓) ∈ V

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
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