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Theorem List for Metamath Proof Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfnelfp 6601 Property of a fixed point of a function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
((𝐹 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹 ∩ I ) ↔ (𝐹𝑋) = 𝑋))

Theoremfndifnfp 6602* Express the class of non-fixed points of a function. (Contributed by Stefan O'Rear, 14-Aug-2015.)
(𝐹 Fn 𝐴 → dom (𝐹 ∖ I ) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ 𝑥})

Theoremfnelnfp 6603 Property of a non-fixed point of a function. (Contributed by Stefan O'Rear, 15-Aug-2015.)
((𝐹 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹 ∖ I ) ↔ (𝐹𝑋) ≠ 𝑋))

Theoremfnnfpeq0 6604 A function is the identity iff it moves no points. (Contributed by Stefan O'Rear, 25-Aug-2015.)
(𝐹 Fn 𝐴 → (dom (𝐹 ∖ I ) = ∅ ↔ 𝐹 = ( I ↾ 𝐴)))

Theoremfvunsn 6605 Remove an ordered pair not participating in a function value. (Contributed by NM, 1-Oct-2013.) (Revised by Mario Carneiro, 28-May-2014.)
(𝐵𝐷 → ((𝐴 ∪ {⟨𝐵, 𝐶⟩})‘𝐷) = (𝐴𝐷))

Theoremfvsn 6606 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.)
𝐴 ∈ V    &   𝐵 ∈ V       ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵

Theoremfvsng 6607 The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.)
((𝐴𝑉𝐵𝑊) → ({⟨𝐴, 𝐵⟩}‘𝐴) = 𝐵)

Theoremfvsnun1 6608 The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 6609. (Contributed by NM, 23-Sep-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))       (𝐺𝐴) = 𝐵

Theoremfvsnun2 6609 The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 6608. (Contributed by NM, 23-Sep-2007.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐺 = ({⟨𝐴, 𝐵⟩} ∪ (𝐹 ↾ (𝐶 ∖ {𝐴})))       (𝐷 ∈ (𝐶 ∖ {𝐴}) → (𝐺𝐷) = (𝐹𝐷))

Theoremfnsnsplit 6610 Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.)
((𝐹 Fn 𝐴𝑋𝐴) → 𝐹 = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}))

Theoremfsnunf 6611 Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
((𝐹:𝑆𝑇 ∧ (𝑋𝑉 ∧ ¬ 𝑋𝑆) ∧ 𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇)

Theoremfsnunf2 6612 Adjoining a point to a punctured function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
((𝐹:(𝑆 ∖ {𝑋})⟶𝑇𝑋𝑆𝑌𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):𝑆𝑇)

Theoremfsnunfv 6613 Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.)
((𝑋𝑉𝑌𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩})‘𝑋) = 𝑌)

Theoremfsnunres 6614 Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
((𝐹 Fn 𝑆 ∧ ¬ 𝑋𝑆) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) ↾ 𝑆) = 𝐹)

Theoremfunresdfunsn 6615 Restricting a function to a domain without one element of the domain of the function, and adding a pair of this element and the function value of the element results in the function itself. (Contributed by AV, 2-Dec-2018.)
((Fun 𝐹𝑋 ∈ dom 𝐹) → ((𝐹 ↾ (V ∖ {𝑋})) ∪ {⟨𝑋, (𝐹𝑋)⟩}) = 𝐹)

Theoremfvpr1 6616 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
𝐴 ∈ V    &   𝐶 ∈ V       (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)

Theoremfvpr2 6617 The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
𝐵 ∈ V    &   𝐷 ∈ V       (𝐴𝐵 → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷)

Theoremfvpr1g 6618 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
((𝐴𝑉𝐶𝑊𝐴𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐴) = 𝐶)

Theoremfvpr2g 6619 The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
((𝐵𝑉𝐷𝑊𝐴𝐵) → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}‘𝐵) = 𝐷)

Theoremfvtp1 6620 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 6621 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 6622 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)

Theoremfvtp1g 6623 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 6624 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 6625 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)

Theoremtpres 6626 An unordered triple of ordered pairs restricted to all but one first components of the pairs is an unordered pair of ordered pairs. (Contributed by AV, 14-Mar-2020.)
(𝜑𝑇 = {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐸𝑉)    &   (𝜑𝐹𝑉)    &   (𝜑𝐵𝐴)    &   (𝜑𝐶𝐴)       (𝜑 → (𝑇 ↾ (V ∖ {𝐴})) = {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})

Theoremfvconst2g 6627 The value of a constant function. (Contributed by NM, 20-Aug-2005.)
((𝐵𝐷𝐶𝐴) → ((𝐴 × {𝐵})‘𝐶) = 𝐵)

Theoremfconst2g 6628 A constant function expressed as a Cartesian product. (Contributed by NM, 27-Nov-2007.)
(𝐵𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))

Theoremfvconst2 6629 The value of a constant function. (Contributed by NM, 16-Apr-2005.)
𝐵 ∈ V       (𝐶𝐴 → ((𝐴 × {𝐵})‘𝐶) = 𝐵)

Theoremfconst2 6630 A constant function expressed as a Cartesian product. (Contributed by NM, 20-Aug-1999.)
𝐵 ∈ V       (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))

Theoremfconst5 6631 Two ways to express that a function is constant. (Contributed by NM, 27-Nov-2007.)
((𝐹 Fn 𝐴𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))

Theoremfnprb 6632 A function whose domain has at most two elements can be represented as a set of at most two ordered pairs. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.) Revised to eliminate unnecessary antecedent 𝐴𝐵. (Revised by NM, 29-Dec-2018.)
𝐴 ∈ V    &   𝐵 ∈ V       (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})

Theoremfntpb 6633 A function whose domain has at most three elements can be represented as a set of at most three ordered pairs. (Contributed by AV, 26-Jan-2021.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝐶 ∈ V       (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩, ⟨𝐶, (𝐹𝐶)⟩})

Theoremfnpr2g 6634 A function whose domain has at most two elements can be represented as a set of at most two ordered pairs. (Contributed by Thierry Arnoux, 12-Jul-2020.)
((𝐴𝑉𝐵𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))

Theoremfpr2g 6635 A function that maps a pair to a class is a pair of ordered pairs. (Contributed by Thierry Arnoux, 12-Jul-2020.)
((𝐴𝑉𝐵𝑊) → (𝐹:{𝐴, 𝐵}⟶𝐶 ↔ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})))

Theoremfconstfv 6636* A constant function expressed in terms of its functionality, domain, and value. See also fconst2 6630. (Contributed by NM, 27-Aug-2004.) (Proof shortened by OpenAI, 25-Mar-2020.)
(𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) = 𝐵))

Theoremfconst3 6637 Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.)
(𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴𝐴 ⊆ (𝐹 “ {𝐵})))

Theoremfconst4 6638 Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
(𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (𝐹 “ {𝐵}) = 𝐴))

Theoremresfunexg 6639 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)

Theoremresiexd 6640 The restriction of the identity relation to a set is a set. (Contributed by AV, 15-Feb-2020.)
(𝜑𝐵𝑉)       (𝜑 → ( I ↾ 𝐵) ∈ V)

Theoremfnex 6641 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 6639. See fnexALT 7293 for alternate proof. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
((𝐹 Fn 𝐴𝐴𝐵) → 𝐹 ∈ V)

Theoremfunex 6642 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 6641. (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 6643* 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 6644* 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 6645 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 6646* If the domain of a function given by maps-to notation is a set, the function is a set. Inference version of mptexg 6644. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)
𝐴 ∈ V       (𝑥𝐴𝐵) ∈ V

Theoremmptexd 6647* If the domain of a function given by maps-to notation is a set, the function is a set. Deduction version of mptexg 6644. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
(𝜑𝐴𝑉)       (𝜑 → (𝑥𝐴𝐵) ∈ V)

Theoremmptrabex 6648* 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 6649 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 6650* 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 6651 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 6652 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)))

Theoremresfvresima 6653 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 6654 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 6655 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 6656* 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 6657* 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 6658* TODO: This is the same as issref 5663 (which has a much longer proof). Should we replace issref 5663 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)

Theoremfvclss 6659* Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.)
{𝑦 ∣ ∃𝑥 𝑦 = (𝐹𝑥)} ⊆ (ran 𝐹 ∪ {∅})

Theoremelabrex 6660* Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
𝐵 ∈ V       (𝑥𝐴𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵})

Theoremabrexco 6661* 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 6662* 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 6663* 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 6664* 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 6665* 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 6666* The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 6665 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 6667* Membership in the union of an image of a function. (Contributed by NM, 28-Sep-2006.)
(Fun 𝐹 → (𝐵 (𝐹𝐴) ↔ ∃𝑥𝐴 𝐵 ∈ (𝐹𝑥)))

Theoremelunirn 6668* Membership in the union of the range of a function. See elunirnALT 6669 for a shorter proof which uses ax-pow 4988. (Contributed by NM, 24-Sep-2006.)
(Fun 𝐹 → (𝐴 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))

TheoremelunirnALT 6669* Alternate proof of elunirn 6668. It is shorter but requires ax-pow 4988 (through eluniima 6667, funiunfv 6665, ndmfv 6375). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(Fun 𝐹 → (𝐴 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))

Theoremfnunirn 6670* 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 6671* 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 6672* 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 6673 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 6674 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 6675 Alternate proof of f1cofveqaeq 6674, 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 6676 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 6677* 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 6678 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 6679 Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐹:𝐴1-1𝐵𝑋𝐴𝑌𝐴) → ((𝐹𝑋) ∈ (𝐹𝑌) ↔ 𝑋𝑌))

Theoremf1imass 6680 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 6681 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 6682 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 6683 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 6684 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 6685* 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 6686* 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 6687* 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 6688 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 6689 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 6690* 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 6691 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 6692 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 6693 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 6694 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 6695 An involution is a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1-onto𝐴)

Theoremnvocnv 6696* 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 6697* 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 6698* 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 6699 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 6700 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))

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