P. 68 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 6701-6800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fnimaeq0 6701	Images under a function never map nonempty sets to empty sets. EDITORIAL: usable in fnwe2lem2 43039. (Contributed by Stefan O'Rear, 21-Jan-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝐹 “ 𝐵) = ∅ ↔ 𝐵 = ∅))
Theorem	dfmpt3 6702	Alternate definition for the maps-to notation df-mpt 5231. (Contributed by Mario Carneiro, 30-Dec-2016.)
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵})
Theorem	mptfnf 6703	The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Thierry Arnoux, 10-May-2017.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
Theorem	fnmptf 6704	The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
Theorem	fnopabg 6705*	Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ↔ 𝐹 Fn 𝐴)
Theorem	fnopab 6706*	Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 5-Mar-1996.)
⊢ (𝑥 ∈ 𝐴 → ∃!𝑦𝜑)    &   ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}    ⇒   ⊢ 𝐹 Fn 𝐴
Theorem	mptfng 6707*	The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴)
Theorem	fnmpt 6708*	The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴)
Theorem	fnmptd 6709*	The maps-to notation defines a function with domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ Ⅎ𝑥𝜑    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝜑 → 𝐹 Fn 𝐴)
Theorem	mpt0 6710	A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ (𝑥 ∈ ∅ ↦ 𝐴) = ∅
Theorem	fnmpti 6711*	Functionality and domain of an ordered-pair class abstraction. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ 𝐹 Fn 𝐴
Theorem	dmmpti 6712*	Domain of the mapping operation. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ dom 𝐹 = 𝐴
Theorem	dmmptd 6713*	The domain of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ 𝐴 = (𝑥 ∈ 𝐵 ↦ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → dom 𝐴 = 𝐵)
Theorem	mptun 6714	Union of mappings which are mutually compatible. (Contributed by Mario Carneiro, 31-Aug-2015.)
⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↦ 𝐶) = ((𝑥 ∈ 𝐴 ↦ 𝐶) ∪ (𝑥 ∈ 𝐵 ↦ 𝐶))
Theorem	partfun 6715	Rewrite a function defined by parts, using a mapping and an if construct, into a union of functions on disjoint domains. (Contributed by Thierry Arnoux, 30-Mar-2017.)
⊢ (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐵, 𝐶, 𝐷)) = ((𝑥 ∈ (𝐴 ∩ 𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴 ∖ 𝐵) ↦ 𝐷))
Theorem	feq1 6716	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵))
Theorem	feq2 6717	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐴 = 𝐵 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶))
Theorem	feq3 6718	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵))
Theorem	feq23 6719	Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷))
Theorem	feq1d 6720	Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
⊢ (𝜑 → 𝐹 = 𝐺)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵))
Theorem	feq1dd 6721	Equality deduction for functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)
Theorem	feq2d 6722	Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶))
Theorem	feq3d 6723	Equality deduction for functions. (Contributed by AV, 1-Jan-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹:𝑋⟶𝐴 ↔ 𝐹:𝑋⟶𝐵))
Theorem	feq12d 6724	Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶))
Theorem	feq123d 6725	Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐷))
Theorem	feq123 6726	Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐶⟶𝐷))
Theorem	feq1i 6727	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐹 = 𝐺    ⇒   ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵)
Theorem	feq2i 6728	Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶)
Theorem	feq12i 6729	Equality inference for functions. (Contributed by AV, 7-Feb-2021.)
⊢ 𝐹 = 𝐺    &   ⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)
Theorem	feq23i 6730	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)
Theorem	feq23d 6731	Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷))
Theorem	nff 6732	Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵
Theorem	sbcfng 6733*	Distribute proper substitution through the function predicate with a domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹 Fn 𝐴 ↔ ⦋𝑋 / 𝑥⦌𝐹 Fn ⦋𝑋 / 𝑥⦌𝐴))
Theorem	sbcfg 6734*	Distribute proper substitution through the function predicate with domain and codomain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
⊢ (𝑋 ∈ 𝑉 → ([𝑋 / 𝑥]𝐹:𝐴⟶𝐵 ↔ ⦋𝑋 / 𝑥⦌𝐹:⦋𝑋 / 𝑥⦌𝐴⟶⦋𝑋 / 𝑥⦌𝐵))
Theorem	elimf 6735	Eliminate a mapping hypothesis for the weak deduction theorem dedth 4588, when a special case 𝐺:𝐴⟶𝐵 is provable, in order to convert 𝐹:𝐴⟶𝐵 from a hypothesis to an antecedent. (Contributed by NM, 24-Aug-2006.)
⊢ 𝐺:𝐴⟶𝐵    ⇒   ⊢ if(𝐹:𝐴⟶𝐵, 𝐹, 𝐺):𝐴⟶𝐵
Theorem	ffn 6736	A mapping is a function with domain. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
Theorem	ffnd 6737	A mapping is a function with domain, deduction form. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → 𝐹 Fn 𝐴)
Theorem	dffn2 6738	Any function is a mapping into V. (Contributed by NM, 31-Oct-1995.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V)
Theorem	ffun 6739	A mapping is a function. (Contributed by NM, 3-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
Theorem	ffund 6740	A mapping is a function, deduction version. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → Fun 𝐹)
Theorem	frel 6741	A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹)
Theorem	freld 6742	A mapping is a relation. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → Rel 𝐹)
Theorem	frn 6743	The range of a mapping. (Contributed by NM, 3-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
Theorem	frnd 6744	Deduction form of frn 6743. The range of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
Theorem	fdm 6745	The domain of a mapping. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Wolf Lammen, 29-May-2024.)
⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
Theorem	fdmd 6746	Deduction form of fdm 6745. The domain of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → dom 𝐹 = 𝐴)
Theorem	fdmi 6747	Inference associated with fdm 6745. The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
⊢ 𝐹:𝐴⟶𝐵    ⇒   ⊢ dom 𝐹 = 𝐴
Theorem	dffn3 6748	A function maps to its range. (Contributed by NM, 1-Sep-1999.)
⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
Theorem	ffrn 6749	A function maps to its range. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)
Theorem	ffrnb 6750	Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6749. (Contributed by BJ, 21-Sep-2024.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
Theorem	ffrnbd 6751	A function maps to its range iff the range is a subset of its codomain. Generalization of ffrn 6749. (Contributed by AV, 20-Sep-2024.)
⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶ran 𝐹))
Theorem	fss 6752	Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴⟶𝐶)
Theorem	fssd 6753	Expanding the codomain of a mapping, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
Theorem	fssdmd 6754	Expressing that a class is a subclass of the domain of a function expressed in maps-to notation, deduction form. (Contributed by AV, 21-Aug-2022.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐷 ⊆ dom 𝐹)    ⇒   ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
Theorem	fssdm 6755	Expressing that a class is a subclass of the domain of a function expressed in maps-to notation, semi-deduction form. (Contributed by AV, 21-Aug-2022.)
⊢ 𝐷 ⊆ dom 𝐹    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → 𝐷 ⊆ 𝐴)
Theorem	fimass 6756	The image of a class under a function with domain and codomain is a subset of its codomain. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝑋) ⊆ 𝐵)
Theorem	fimassd 6757	The image of a class is a subset of its codomain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐹 “ 𝑋) ⊆ 𝐵)
Theorem	fimacnv 6758	The preimage of the codomain of a function is the function's domain. (Contributed by FL, 25-Jan-2007.) (Proof shortened by AV, 20-Sep-2024.)
⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴)
Theorem	fcof 6759	Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 6760. (Contributed by AV, 18-Sep-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
Theorem	fco 6760	Composition of two functions with domain and codomain as a function with domain and codomain. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Proof shortened by AV, 20-Sep-2024.)
⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
Theorem	fcod 6761	Composition of two mappings. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐵⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
Theorem	fco2 6762	Functionality of a composition with weakened out of domain condition on the first argument. (Contributed by Stefan O'Rear, 11-Mar-2015.)
⊢ (((𝐹 ↾ 𝐵):𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
Theorem	fssxp 6763	A mapping is a class of ordered pairs. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (𝐹:𝐴⟶𝐵 → 𝐹 ⊆ (𝐴 × 𝐵))
Theorem	funssxp 6764	Two ways of specifying a partial function from 𝐴 to 𝐵. (Contributed by NM, 13-Nov-2007.)
⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ (𝐴 × 𝐵)) ↔ (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))
Theorem	ffdm 6765	A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))
Theorem	ffdmd 6766	The domain of a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → 𝐹:dom 𝐹⟶𝐵)
Theorem	fdmrn 6767	A different way to write 𝐹 is a function. (Contributed by Thierry Arnoux, 7-Dec-2016.)
⊢ (Fun 𝐹 ↔ 𝐹:dom 𝐹⟶ran 𝐹)
Theorem	funcofd 6768	Composition of two functions as a function with domain and codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Proof shortened by AV, 20-Sep-2024.)
⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → Fun 𝐺)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)
Theorem	opelf 6769	The members of an ordered pair element of a mapping belong to the mapping's domain and codomain. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ ⟨𝐶, 𝐷⟩ ∈ 𝐹) → (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))
Theorem	fun 6770	The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐷) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶(𝐶 ∪ 𝐷))
Theorem	fun2 6771	The union of two functions with disjoint domains. (Contributed by Mario Carneiro, 12-Mar-2015.)
⊢ (((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)
Theorem	fun2d 6772	The union of functions with disjoint domains is a function, deduction version of fun2 6771. (Contributed by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝐶)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    ⇒   ⊢ (𝜑 → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)
Theorem	fnfco 6773	Composition of two functions. (Contributed by NM, 22-May-2006.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)
Theorem	fssres 6774	Restriction of a function with a subclass of its domain. (Contributed by NM, 23-Sep-2004.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
Theorem	fssresd 6775	Restriction of a function with a subclass of its domain, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐶 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
Theorem	fssres2 6776	Restriction of a restricted function with a subclass of its domain. (Contributed by NM, 21-Jul-2005.)
⊢ (((𝐹 ↾ 𝐴):𝐴⟶𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
Theorem	fresin 6777	An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)
⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵)
Theorem	resasplit 6778	If two functions agree on their common domain, express their union as a union of three functions with pairwise disjoint domains. (Contributed by Stefan O'Rear, 9-Oct-2014.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺) = ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))))
Theorem	fresaun 6779	The union of two functions which agree on their common domain is a function. (Contributed by Stefan O'Rear, 9-Oct-2014.)
⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶𝐶)
Theorem	fresaunres2 6780	From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Stefan O'Rear, 9-Oct-2014.)
⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)
Theorem	fresaunres1 6781	From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Mario Carneiro, 16-Feb-2015.)
⊢ ((𝐹:𝐴⟶𝐶 ∧ 𝐺:𝐵⟶𝐶 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹)
Theorem	fcoi1 6782	Composition of a mapping and restricted identity. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹)
Theorem	fcoi2 6783	Composition of restricted identity and a mapping. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (𝐹:𝐴⟶𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹)
Theorem	feu 6784*	There is exactly one value of a function in its codomain. (Contributed by NM, 10-Dec-2003.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 ⟨𝐶, 𝑦⟩ ∈ 𝐹)
Theorem	fcnvres 6785	The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
⊢ (𝐹:𝐴⟶𝐵 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ 𝐵))
Theorem	fimacnvdisj 6786	The preimage of a class disjoint with a mapping's codomain is empty. (Contributed by FL, 24-Jan-2007.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (◡𝐹 “ 𝐶) = ∅)
Theorem	fint 6787*	Function into an intersection. (Contributed by NM, 14-Oct-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ 𝐵 ≠ ∅    ⇒   ⊢ (𝐹:𝐴⟶∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐹:𝐴⟶𝑥)
Theorem	fin 6788	Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (𝐹:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝐹:𝐴⟶𝐵 ∧ 𝐹:𝐴⟶𝐶))
Theorem	f0 6789	The empty function. (Contributed by NM, 14-Aug-1999.)
⊢ ∅:∅⟶𝐴
Theorem	f00 6790	A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.)
⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Theorem	f0bi 6791	A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅)
Theorem	f0dom0 6792	A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.)
⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅))
Theorem	f0rn0 6793*	If there is no element in the range of a function, its domain must be empty. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
⊢ ((𝐸:𝑋⟶𝑌 ∧ ¬ ∃𝑦 ∈ 𝑌 𝑦 ∈ ran 𝐸) → 𝑋 = ∅)
Theorem	fconst 6794	A Cartesian product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 × {𝐵}):𝐴⟶{𝐵}
Theorem	fconstg 6795	A Cartesian product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
Theorem	fnconstg 6796	A Cartesian product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}) Fn 𝐴)
Theorem	fconst6g 6797	Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶𝐶)
Theorem	fconst6 6798	A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)
⊢ 𝐵 ∈ 𝐶    ⇒   ⊢ (𝐴 × {𝐵}):𝐴⟶𝐶
Theorem	f1eq1 6799	Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐵 ↔ 𝐺:𝐴–1-1→𝐵))
Theorem	f1eq2 6800	Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐹:𝐵–1-1→𝐶))