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Theorem List for Intuitionistic Logic Explorer - 5401-5500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfvtp1g 5401 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(((𝐴𝑉𝐷𝑊) ∧ (𝐴𝐵𝐴𝐶)) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐴) = 𝐷)
 
Theoremfvtp2g 5402 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(((𝐵𝑉𝐸𝑊) ∧ (𝐴𝐵𝐵𝐶)) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐵) = 𝐸)
 
Theoremfvtp3g 5403 The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
(((𝐶𝑉𝐹𝑊) ∧ (𝐴𝐶𝐵𝐶)) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = 𝐹)
 
Theoremfvtp1 5404 The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
𝐴 ∈ V    &   𝐷 ∈ V       ((𝐴𝐵𝐴𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐴) = 𝐷)
 
Theoremfvtp2 5405 The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
𝐵 ∈ V    &   𝐸 ∈ V       ((𝐴𝐵𝐵𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐵) = 𝐸)
 
Theoremfvtp3 5406 The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
𝐶 ∈ V    &   𝐹 ∈ V       ((𝐴𝐶𝐵𝐶) → ({⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}‘𝐶) = 𝐹)
 
Theoremfvconst2g 5407 The value of a constant function. (Contributed by NM, 20-Aug-2005.)
((𝐵𝐷𝐶𝐴) → ((𝐴 × {𝐵})‘𝐶) = 𝐵)
 
Theoremfconst2g 5408 A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.)
(𝐵𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
 
Theoremfvconst2 5409 The value of a constant function. (Contributed by NM, 16-Apr-2005.)
𝐵 ∈ V       (𝐶𝐴 → ((𝐴 × {𝐵})‘𝐶) = 𝐵)
 
Theoremfconst2 5410 A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.)
𝐵 ∈ V       (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))
 
Theoremfconstfvm 5411* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5410. (Contributed by Jim Kingdon, 8-Jan-2019.)
(∃𝑦 𝑦𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵)))
 
Theoremfconst3m 5412* Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.)
(∃𝑥 𝑥𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵}))))
 
Theoremfconst4m 5413* Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
(∃𝑥 𝑥𝐴 → (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴)))
 
Theoremresfunexg 5414 The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.)
((Fun 𝐴𝐵𝐶) → (𝐴𝐵) ∈ V)
 
Theoremfnex 5415 If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 5414. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
 
Theoremfunex 5416 If the domain of a function exists, so does the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 5415. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.)
((Fun 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)
 
Theoremopabex 5417* Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)
𝐴 ∈ V    &   (𝑥𝐴 → ∃*𝑦𝜑)       {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)} ∈ V
 
Theoremmptexg 5418* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
 
Theoremmptex 5419* If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)
𝐴 ∈ V       (𝑥𝐴𝐵) ∈ V
 
Theoremfex 5420 If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)
((𝐹:𝐴𝐵𝐴𝐶) → 𝐹 ∈ V)
 
Theoremeufnfv 5421* A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)
𝐴 ∈ V    &   𝐵 ∈ V       ∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑓𝑥) = 𝐵)
 
Theoremfunfvima 5422 A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)
((Fun 𝐹𝐵 ∈ dom 𝐹) → (𝐵𝐴 → (𝐹𝐵) ∈ (𝐹𝐴)))
 
Theoremfunfvima2 5423 A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐵𝐴 → (𝐹𝐵) ∈ (𝐹𝐴)))
 
Theoremfunfvima3 5424 A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.)
((Fun 𝐹𝐹𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹𝐴) ∈ (𝐺 “ {𝐴})))
 
Theoremfnfvima 5425 The function value of an operand in a set is contained in the image of that set, using the Fn abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.)
((𝐹 Fn 𝐴𝑆𝐴𝑋𝑆) → (𝐹𝑋) ∈ (𝐹𝑆))
 
Theoremrexima 5426* Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
(𝑥 = (𝐹𝑦) → (𝜑𝜓))       ((𝐹 Fn 𝐴𝐵𝐴) → (∃𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∃𝑦𝐵 𝜓))
 
Theoremralima 5427* Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.)
(𝑥 = (𝐹𝑦) → (𝜑𝜓))       ((𝐹 Fn 𝐴𝐵𝐴) → (∀𝑥 ∈ (𝐹𝐵)𝜑 ↔ ∀𝑦𝐵 𝜓))
 
Theoremidref 5428* TODO: This is the same as issref 4737 (which has a much longer proof). Should we replace issref 4737 with this one? - NM 9-May-2016.

Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.)

(( I ↾ 𝐴) ⊆ 𝑅 ↔ ∀𝑥𝐴 𝑥𝑅𝑥)
 
Theoremelabrex 5429* Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
𝐵 ∈ V       (𝑥𝐴𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
 
Theoremabrexco 5430* Composition of two image maps 𝐶(𝑦) and 𝐵(𝑤). (Contributed by NM, 27-May-2013.)
𝐵 ∈ V    &   (𝑦 = 𝐵𝐶 = 𝐷)       {𝑥 ∣ ∃𝑦 ∈ {𝑧 ∣ ∃𝑤𝐴 𝑧 = 𝐵}𝑥 = 𝐶} = {𝑥 ∣ ∃𝑤𝐴 𝑥 = 𝐷}
 
Theoremimaiun 5431* The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)
(𝐴 𝑥𝐵 𝐶) = 𝑥𝐵 (𝐴𝐶)
 
Theoremimauni 5432* The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)
(𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥)
 
Theoremfniunfv 5433* The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.)
(𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
 
Theoremfuniunfvdm 5434* The indexed union of a function's values is the union of its image under the index class. This theorem is a slight variation of fniunfv 5433. (Contributed by Jim Kingdon, 10-Jan-2019.)
(𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
 
Theoremfuniunfvdmf 5435* The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5434 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.)
𝑥𝐹       (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
 
Theoremeluniimadm 5436* Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.)
(𝐹 Fn 𝐴 → (𝐵 (𝐹𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ (𝐹𝑥)))
 
Theoremelunirn 5437* Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.)
(Fun 𝐹 → (𝐴 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
 
Theoremfnunirn 5438* Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝐹 Fn 𝐼 → (𝐴 ran 𝐹 ↔ ∃𝑥𝐼 𝐴 ∈ (𝐹𝑥)))
 
Theoremdff13 5439* A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.)
(𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
 
Theoremf1veqaeq 5440 If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)
((𝐹:𝐴1-1𝐵 ∧ (𝐶𝐴𝐷𝐴)) → ((𝐹𝐶) = (𝐹𝐷) → 𝐶 = 𝐷))
 
Theoremdff13f 5441* A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.)
𝑥𝐹    &   𝑦𝐹       (𝐹:𝐴1-1𝐵 ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
 
Theoremf1mpt 5442* 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 5443 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 5444 Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ 𝑋𝑌))
 
Theoremf1imass 5445 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 5446 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))
 
Theoremdff1o6 5447* 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 5448 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 5449 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 5450 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 5451 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))
 
Theoremf1ocnvdm 5452 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 5453 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 5454 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 5455 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 5456* 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 5457* 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 5458 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 5459 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))
 
Theoremfcof1o 5460 Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.)
(((𝐹:𝐴𝐵𝐺:𝐵𝐴) ∧ ((𝐹𝐺) = ( I ↾ 𝐵) ∧ (𝐺𝐹) = ( I ↾ 𝐴))) → (𝐹:𝐴1-1-onto𝐵𝐹 = 𝐺))
 
Theoremfoeqcnvco 5461 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 5462 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)))
 
Theoremfliftrel 5463* 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 5464* 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 5465* Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑅)    &   ((𝜑𝑥𝑋) → 𝐵𝑆)       ((𝜑𝑥𝑋) → 𝐴𝐹𝐵)
 
Theoremfliftcnv 5466* 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 5467* 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 5468* 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 5469* 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 5470* 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 5471* 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 5472 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 5473 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 5474 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 5475 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 5476 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 5477 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 5478 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)
 
Theoremisorel 5479 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)))
 
Theoremisoresbr 5480* A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.)
((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))
 
Theoremisoid 5481 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 5482 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 5483 Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆(𝐴, 𝐵))
 
Theoremisores2 5484 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 5485 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 5486 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 5487 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))
 
Theoremisoini 5488 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 5489 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))
 
Theoremisoselem 5490* Lemma for isose 5491. (Contributed by Mario Carneiro, 23-Jun-2015.)
(𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑 → (𝐻𝑥) ∈ V)       (𝜑 → (𝑅 Se 𝐴𝑆 Se 𝐵))
 
Theoremisose 5491 An isomorphism preserves set-like relations. (Contributed by Mario Carneiro, 23-Jun-2015.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Se 𝐴𝑆 Se 𝐵))
 
Theoremisopolem 5492 Lemma for isopo 5493. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Po 𝐵𝑅 Po 𝐴))
 
Theoremisopo 5493 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 5494 Lemma for isoso 5495. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑆 Or 𝐵𝑅 Or 𝐴))
 
Theoremisoso 5495 An isomorphism preserves strict ordering. (Contributed by Stefan O'Rear, 16-Nov-2014.)
(𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴𝑆 Or 𝐵))
 
Theoremf1oiso 5496* 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 5497* 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))
 
2.6.9  Restricted iota (description binder)
 
Syntaxcrio 5498 Extend class notation with restricted description binder.
class (𝑥𝐴 𝜑)
 
Definitiondf-riota 5499 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 4897. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 2-Sep-2018.)
(𝑥𝐴 𝜑) = (℩𝑥(𝑥𝐴𝜑))
 
Theoremriotaeqdv 5500* Formula-building deduction rule for iota. (Contributed by NM, 15-Sep-2011.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐵 𝜓))
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