P. 73 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 7201-7300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fvconst2g 7201	The value of a constant function. (Contributed by NM, 20-Aug-2005.)
⊢ ((𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝐶) = 𝐵)
Theorem	fconst2g 7202	A constant function expressed as a Cartesian product. (Contributed by NM, 27-Nov-2007.)
⊢ (𝐵 ∈ 𝐶 → (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵})))
Theorem	fvconst2 7203	The value of a constant function. (Contributed by NM, 16-Apr-2005.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐶 ∈ 𝐴 → ((𝐴 × {𝐵})‘𝐶) = 𝐵)
Theorem	fconst2 7204	A constant function expressed as a Cartesian product. (Contributed by NM, 20-Aug-1999.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐹:𝐴⟶{𝐵} ↔ 𝐹 = (𝐴 × {𝐵}))
Theorem	fconst5 7205	Two ways to express that a function is constant. (Contributed by NM, 27-Nov-2007.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ≠ ∅) → (𝐹 = (𝐴 × {𝐵}) ↔ ran 𝐹 = {𝐵}))
Theorem	rnmptc 7206*	Range of a constant function in maps-to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Remove extra hypothesis. (Revised by SN, 17-Apr-2024.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐴 ≠ ∅)    ⇒   ⊢ (𝜑 → ran 𝐹 = {𝐵})
Theorem	fnprb 7207	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.) Eliminate unnecessary antecedent 𝐴 ≠ 𝐵. (Revised by NM, 29-Dec-2018.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})
Theorem	fntpb 7208	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, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})
Theorem	fnpr2g 7209	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.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}))
Theorem	fpr2g 7210	A function that maps a pair to a class is a pair of ordered pairs. (Contributed by Thierry Arnoux, 12-Jul-2020.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹:{𝐴, 𝐵}⟶𝐶 ↔ ((𝐹‘𝐴) ∈ 𝐶 ∧ (𝐹‘𝐵) ∈ 𝐶 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})))
Theorem	fconstfv 7211*	A constant function expressed in terms of its functionality, domain, and value. See also fconst2 7204. (Contributed by NM, 27-Aug-2004.) (Proof shortened by OpenAI, 25-Mar-2020.)
⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
Theorem	fconst3 7212	Two ways to express a constant function. (Contributed by NM, 15-Mar-2007.)
⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
Theorem	fconst4 7213	Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.)
⊢ (𝐹:𝐴⟶{𝐵} ↔ (𝐹 Fn 𝐴 ∧ (◡𝐹 “ {𝐵}) = 𝐴))
Theorem	resfunexg 7214	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 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 ↾ 𝐵) ∈ V)
Theorem	resiexd 7215	The restriction of the identity relation to a set is a set. (Contributed by AV, 15-Feb-2020.)
⊢ (𝜑 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ( I ↾ 𝐵) ∈ V)
Theorem	fnex 7216	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 7214. See fnexALT 7948 for alternate proof. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V)
Theorem	fnexd 7217	If the domain of a function is a set, the function is a set. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 ∈ V)
Theorem	funex 7218	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 7216. (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 𝐹 ∈ 𝐵) → 𝐹 ∈ V)
Theorem	opabex 7219*	Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.)
⊢ 𝐴 ∈ V    &   ⊢ (𝑥 ∈ 𝐴 → ∃*𝑦𝜑)    ⇒   ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V
Theorem	mptexg 7220*	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.)
⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
Theorem	mptexgf 7221	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.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
Theorem	mptex 7222*	If the domain of a function given by maps-to notation is a set, the function is a set. Inference version of mptexg 7220. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V
Theorem	mptexd 7223*	If the domain of a function given by maps-to notation is a set, the function is a set. Deduction version of mptexg 7220. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V)
Theorem	mptrabex 7224*	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 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↦ 𝐵) ∈ V
Theorem	fex 7225	If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝐶) → 𝐹 ∈ V)
Theorem	fexd 7226	If the domain of a mapping is a set, the function is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐹 ∈ V)
Theorem	mptfvmpt 7227*	A function in maps-to notation as the value of another function in maps-to notation. (Contributed by AV, 20-Aug-2022.)
⊢ (𝑦 = 𝑌 → 𝑀 = (𝑥 ∈ 𝑉 ↦ 𝐴))    &   ⊢ 𝐺 = (𝑦 ∈ 𝑊 ↦ 𝑀)    &   ⊢ 𝑉 = (𝐹‘𝑋)    ⇒   ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴))
Theorem	eufnfv 7228*	A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ∃!𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) = 𝐵)
Theorem	funfvima 7229	A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.)
⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴)))
Theorem	funfvima2 7230	A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴)))
Theorem	funfvima2d 7231	A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by AV, 23-Mar-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴))
Theorem	fnfvima 7232	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 𝐴 ∧ 𝑆 ⊆ 𝐴 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆))
Theorem	fnfvimad 7233	A function's value belongs to the image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐶))
Theorem	resfvresima 7234	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)    &   ⊢ (𝜑 → 𝑆 ⊆ dom 𝐹)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘((𝐹 ↾ 𝑆)‘𝑋)) = (𝐻‘(𝐹‘𝑋)))
Theorem	funfvima3 7235	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 𝐹 ∧ 𝐹 ⊆ 𝐺) → (𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) ∈ (𝐺 “ {𝐴})))
Theorem	ralima 7236*	Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) Reduce DV conditions. (Revised by Matthew House, 14-Aug-2025.)
⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐵 𝜓))
Theorem	rexima 7237*	Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) Reduce DV conditions. (Revised by Matthew House, 14-Aug-2025.)
⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑥 ∈ (𝐹 “ 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐵 𝜓))
Theorem	fvclss 7238*	Upper bound for the class of values of a class. (Contributed by NM, 9-Nov-1995.)
⊢ {𝑦 ∣ ∃𝑥 𝑦 = (𝐹‘𝑥)} ⊆ (ran 𝐹 ∪ {∅})
Theorem	elabrex 7239*	Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
Theorem	elabrexg 7240*	Elementhood in an image set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
Theorem	abrexco 7241*	Composition of two image maps 𝐶(𝑦) and 𝐵(𝑤). (Contributed by NM, 27-May-2013.)
⊢ 𝐵 ∈ V    &   ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷)    ⇒   ⊢ {𝑥 ∣ ∃𝑦 ∈ {𝑧 ∣ ∃𝑤 ∈ 𝐴 𝑧 = 𝐵}𝑥 = 𝐶} = {𝑥 ∣ ∃𝑤 ∈ 𝐴 𝑥 = 𝐷}
Theorem	imaiun 7242*	The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)
⊢ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝐶)
Theorem	imauni 7243*	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) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝑥)
Theorem	fniunfv 7244*	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 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹)
Theorem	funiunfv 7245*	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 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴))
Theorem	funiunfvf 7246*	The indexed union of a function's values is the union of its image under the index class. This version of funiunfv 7245 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.)
⊢ Ⅎ𝑥𝐹    ⇒   ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴))
Theorem	eluniima 7247*	Membership in the union of an image of a function. (Contributed by NM, 28-Sep-2006.)
⊢ (Fun 𝐹 → (𝐵 ∈ ∪ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝐵 ∈ (𝐹‘𝑥)))
Theorem	elunirn 7248*	Membership in the union of the range of a function. See elunirnALT 7249 for a shorter proof which uses ax-pow 5330. See elfvunirn 6908 for a more general version of the reverse direction. (Contributed by NM, 24-Sep-2006.)
⊢ (Fun 𝐹 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
Theorem	elunirnALT 7249*	Alternate proof of elunirn 7248. It is shorter but requires ax-pow 5330 (through eluniima 7247, funiunfv 7245, ndmfv 6910). (Contributed by NM, 24-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (Fun 𝐹 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
Theorem	fnunirn 7250*	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 ↔ ∃𝑥 ∈ 𝐼 𝐴 ∈ (𝐹‘𝑥)))
Theorem	dff13 7251*	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:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
Theorem	dff13f 7252*	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.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑦𝐹    ⇒   ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
Theorem	f1veqaeq 7253	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:𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))
Theorem	f1cofveqaeq 7254	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:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → ((𝐹‘(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑌)) → 𝑋 = 𝑌))
Theorem	f1cofveqaeqALT 7255	Alternate proof of f1cofveqaeq 7254, 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	dff14i 7256	A one-to-one function maps different arguments onto different values. Implication of the alternate definition dff14a 7267. (Contributed by AV, 30-Oct-2025.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (𝐹‘𝑋) ≠ (𝐹‘𝑌))
Theorem	2f1fvneq 7257	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.) (Proof shortened by AV, 30-Oct-2025.)
⊢ (((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 7258*	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 7259	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 7260	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 7261	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 7262	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 7263	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 7264	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 7265	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 7266*	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 7267*	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 7268*	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	dff15 7269*	A one-to-one function in terms of different arguments never having the same function value. (Contributed by BTernaryTau, 24-Oct-2023.)
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ¬ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥 ≠ 𝑦)))
Theorem	f1resveqaeq 7270	If a function restricted to a class is one-to-one, then for any two elements of the class, the values of the function at those elements are equal only if the two elements are the same element. (Contributed by BTernaryTau, 27-Sep-2023.)
⊢ (((𝐹 ↾ 𝐴):𝐴–1-1→𝐵 ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷))
Theorem	f1resrcmplf1dlem 7271	Lemma for f1resrcmplf1d 7272. (Contributed by BTernaryTau, 27-Sep-2023.) (Revised by Mingli Yuan, 15-Aug-2026.)
⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐷 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ 𝐷)) = ∅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐶)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐷)    &   ⊢ (𝜑 → (𝐹‘𝑋) = (𝐹‘𝑌))    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	f1resrcmplf1d 7272	If a function's restriction to a subclass of its domain and its restriction to the relative complement of that subclass are both one-to-one, and if the ranges of those two restrictions are disjoint, then the function is itself one-to-one. (Contributed by BTernaryTau, 28-Sep-2023.)
⊢ (𝜑 → 𝐶 ⊆ 𝐴)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)    &   ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ 𝐶)):(𝐴 ∖ 𝐶)–1-1→𝐵)    &   ⊢ (𝜑 → ((𝐹 “ 𝐶) ∩ (𝐹 “ (𝐴 ∖ 𝐶))) = ∅)    ⇒   ⊢ (𝜑 → 𝐹:𝐴–1-1→𝐵)
Theorem	f1ounsn 7273	Extension of a bijection by an ordered pair. (Contributed by AV, 15-Sep-2025.)
⊢ 𝐹 = (𝐺 ∪ {⟨𝑋, 𝑌⟩})    ⇒   ⊢ ((𝐺:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) ∧ (𝑋 ∉ 𝐴 ∧ 𝑌 ∉ 𝐵)) → 𝐹:(𝐴 ∪ {𝑋})–1-1-onto→(𝐵 ∪ {𝑌}))
Theorem	f12dfv 7274	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 7275	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 7276*	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 7277	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 7278	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 7279	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 7280	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 7281	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 7282*	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 7283*	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 7284*	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 7285*	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 7286	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 7287	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 7288	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 7289	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 7290*	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 7291*	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 7292	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 7293	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 7294	Show that a function is the inverse of a bijective function if their composition is the identity function. Formerly part of proof of fcof1o 7297. (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 7295	A function is bijective if a "retraction" and a "section" exist, see comments for fcof1 7288 and fcofo 7289. Formerly part of proof of fcof1o 7297. (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 7296	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 7297	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 7298*	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 7299*	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 7300*	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)