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Theorem List for Metamath Proof Explorer - 7101-7200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremresiexd 7101 The restriction of the identity relation to a set is a set. (Contributed by AV, 15-Feb-2020.)
(𝜑𝐵𝑉)       (𝜑 → ( I ↾ 𝐵) ∈ V)
 
Theoremfnex 7102 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 7100. See fnexALT 7802 for alternate proof. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)
 
Theoremfnexd 7103 If the domain of a function is a set, the function is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐴𝑉)       (𝜑𝐹 ∈ V)
 
Theoremfunex 7104 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 7102. (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 7105* 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 7106* 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)
 
Theoremmptexgf 7107 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.) (Revised by Thierry Arnoux, 17-May-2020.)
𝑥𝐴       (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)
 
Theoremmptex 7108* If the domain of a function given by maps-to notation is a set, the function is a set. Inference version of mptexg 7106. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)
𝐴 ∈ V       (𝑥𝐴𝐵) ∈ V
 
Theoremmptexd 7109* If the domain of a function given by maps-to notation is a set, the function is a set. Deduction version of mptexg 7106. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴𝑉)       (𝜑 → (𝑥𝐴𝐵) ∈ V)
 
Theoremmptrabex 7110* If the domain of a function given by maps-to notation is a class abstraction based on a set, the function is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
𝐴 ∈ V       (𝑥 ∈ {𝑦𝐴𝜑} ↦ 𝐵) ∈ V
 
Theoremfex 7111 If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)
((𝐹:𝐴𝐵𝐴𝐶) → 𝐹 ∈ V)
 
Theoremfexd 7112 If the domain of a mapping is a set, the function is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐴𝐶)       (𝜑𝐹 ∈ V)
 
Theoremmptfvmpt 7113* A function in maps-to notation as the value of another function in maps-to notation. (Contributed by AV, 20-Aug-2022.)
(𝑦 = 𝑌𝑀 = (𝑥𝑉𝐴))    &   𝐺 = (𝑦𝑊𝑀)    &   𝑉 = (𝐹𝑋)       (𝑌𝑊 → (𝐺𝑌) = (𝑥𝑉𝐴))
 
Theoremeufnfv 7114* 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 7115 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 7116 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)))
 
Theoremfunfvima2d 7117 A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by AV, 23-Mar-2024.)
(𝜑𝐹:𝐴𝐵)       ((𝜑𝑋𝐴) → (𝐹𝑋) ∈ (𝐹𝐴))
 
Theoremfnfvima 7118 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))
 
Theoremfnfvimad 7119 A function's value belongs to the image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐵𝐴)    &   (𝜑𝐵𝐶)       (𝜑 → (𝐹𝐵) ∈ (𝐹𝐶))
 
Theoremresfvresima 7120 The value of the function value of a restriction for a function restricted to the image of the restricting subset. (Contributed by AV, 6-Mar-2021.)
(𝜑 → Fun 𝐹)    &   (𝜑𝑆 ⊆ dom 𝐹)    &   (𝜑𝑋𝑆)       (𝜑 → ((𝐻 ↾ (𝐹𝑆))‘((𝐹𝑆)‘𝑋)) = (𝐻‘(𝐹𝑋)))
 
Theoremfunfvima3 7121 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})))
 
Theoremrexima 7122* 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 7123* 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))
 
Theoremfvclss 7124* Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.)
{𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ (ran 𝐹 ∪ {∅})
 
Theoremelabrex 7125* Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
𝐵 ∈ V       (𝑥𝐴𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})
 
Theoremabrexco 7126* 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 7127* 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 7128* 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 7129* 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)
 
Theoremfuniunfv 7130* The indexed union of a function's values is the union of its image under the index class.

Note: This theorem depends on the fact that our function value is the empty set outside of its domain. If the antecedent is changed to 𝐹 Fn 𝐴, the theorem can be proved without this dependency. (Contributed by NM, 26-Mar-2006.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

(Fun 𝐹 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
 
Theoremfuniunfvf 7131* The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 7130 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.) (Revised by David Abernethy, 15-Apr-2013.)
𝑥𝐹       (Fun 𝐹 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
 
Theoremeluniima 7132* Membership in the union of an image of a function. (Contributed by NM, 28-Sep-2006.)
(Fun 𝐹 → (𝐵 (𝐹𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ (𝐹𝑥)))
 
Theoremelunirn 7133* Membership in the union of the range of a function. See elunirnALT 7134 for a shorter proof which uses ax-pow 5289. (Contributed by NM, 24-Sep-2006.)
(Fun 𝐹 → (𝐴 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
 
TheoremelunirnALT 7134* Alternate proof of elunirn 7133. It is shorter but requires ax-pow 5289 (through eluniima 7132, funiunfv 7130, ndmfv 6813). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(Fun 𝐹 → (𝐴 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
 
Theoremelunirn2 7135 Condition for the membership in the union of the range of a function. (Contributed by Thierry Arnoux, 13-Nov-2016.)
((Fun 𝐹𝐵 ∈ (𝐹𝐴)) → 𝐵 ran 𝐹)
 
Theoremfnunirn 7136* 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 7137* 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)))
 
Theoremdff13f 7138* 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)))
 
Theoremf1veqaeq 7139 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))
 
Theoremf1cofveqaeq 7140 If the values of a composition of one-to-one functions for two arguments are equal, the arguments themselves must be equal. (Contributed by AV, 3-Feb-2021.)
(((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → 𝑋 = 𝑌))
 
Theoremf1cofveqaeqALT 7141 Alternate proof of f1cofveqaeq 7140, 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:𝐵1-1𝐶𝐺:𝐴1-1𝐵) ∧ (𝑋𝐴𝑌𝐴)) → ((𝐹‘(𝐺𝑋)) = (𝐹‘(𝐺𝑌)) → 𝑋 = 𝑌))
 
Theorem2f1fvneq 7142 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 7143* 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 7144 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 7145 Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ 𝑋𝑌))
 
Theoremf1imass 7146 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 7147 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 7148 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 7149 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 7150 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 7151* 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 7152* 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 7153* 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 7154 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 7155 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 7156* 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 7157 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 7158 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 7159 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 7160 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 7161 An involution is a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1-onto𝐴)
 
Theoremnvocnv 7162* The converse of an involution is the function itself. (Contributed by Thierry Arnoux, 7-May-2019.)
((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = 𝐹)
 
Theoremf1cdmsn 7163* 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:𝐴1-1→{𝐵} ∧ 𝐴 ≠ ∅) → ∃𝑥 𝐴 = {𝑥})
 
Theoremfsnex 7164* 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 7165* 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 7166 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 7167 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 7168 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 7169 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 7170* 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 7171* 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 7172 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 7173 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 7174 Show that a function is the inverse of a bijective function if their composition is the identity function. Formerly part of proof of fcof1o 7177. (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 7175 A function is bijective if a "retraction" and a "section" exist, see comments for fcof1 7168 and fcofo 7169. Formerly part of proof of fcof1o 7177. (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 7176 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 7177 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 7178* 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 7179* 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 7180* 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 7181 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 7182 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:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → (𝐹 = 𝐺 ↔ (𝐹𝐺) = ( I ↾ 𝐴)))
 
Theoremf1eqcocnvOLD 7183 Obsolete version of f1eqcocnv 7182 as of 29-May-2024. (Contributed by Stefan O'Rear, 12-Feb-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐹:𝐴1-1𝐵𝐺:𝐴1-1𝐵) → (𝐹 = 𝐺 ↔ (𝐹𝐺) = ( I ↾ 𝐴)))
 
Theoremfveqf1o 7184 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))
 
Theoremnf1const 7185 A constant function from at least two elements is not one-to-one. (Contributed by AV, 30-Mar-2024.)
((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ 𝐹:𝐴1-1𝐶)
 
Theoremnf1oconst 7186 A constant function from at least two elements is not bijective. (Contributed by AV, 30-Mar-2024.)
((𝐹:𝐴⟶{𝐵} ∧ (𝑋𝐴𝑌𝐴𝑋𝑌)) → ¬ 𝐹:𝐴1-1-onto𝐶)
 
Theoremf1ofvswap 7187 Swapping two values in a bijection between two classes yields another bijection between those classes. (Contributed by BTernaryTau, 31-Aug-2024.)
((𝐹:𝐴1-1-onto𝐵𝑋𝐴𝑌𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹𝑌)⟩, ⟨𝑌, (𝐹𝑋)⟩}):𝐴1-1-onto𝐵)
 
Theoremfliftrel 7188* 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 7189* 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 7190* Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
𝐹 = ran (𝑥𝑋 ↦ ⟨𝐴, 𝐵⟩)    &   ((𝜑𝑥𝑋) → 𝐴𝑅)    &   ((𝜑𝑥𝑋) → 𝐵𝑆)       ((𝜑𝑥𝑋) → 𝐴𝐹𝐵)
 
Theoremfliftcnv 7191* 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 7192* 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 7193* 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 7194* 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 7195* 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 7196* 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 7197 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 7198 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 7199 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 7200 Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)
(𝐴 = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, 𝐵)))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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 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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 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