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	feq1d 6701	Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
⊢ (𝜑 → 𝐹 = 𝐺)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵))
Theorem	feq2d 6702	Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶))
Theorem	feq3d 6703	Equality deduction for functions. (Contributed by AV, 1-Jan-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹:𝑋⟶𝐴 ↔ 𝐹:𝑋⟶𝐵))
Theorem	feq12d 6704	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 6705	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 6706	Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐶⟶𝐷))
Theorem	feq1i 6707	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐹 = 𝐺    ⇒   ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵)
Theorem	feq2i 6708	Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶)
Theorem	feq12i 6709	Equality inference for functions. (Contributed by AV, 7-Feb-2021.)
⊢ 𝐹 = 𝐺    &   ⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐹:𝐴⟶𝐶 ↔ 𝐺:𝐵⟶𝐶)
Theorem	feq23i 6710	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)
Theorem	feq23d 6711	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 6712	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 6713*	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 6714*	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 6715	Eliminate a mapping hypothesis for the weak deduction theorem dedth 4585, 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 6716	A mapping is a function with domain. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
Theorem	ffnd 6717	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 6718	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 6719	A mapping is a function. (Contributed by NM, 3-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
Theorem	ffund 6720	A mapping is a function, deduction version. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → Fun 𝐹)
Theorem	frel 6721	A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹)
Theorem	freld 6722	A mapping is a relation. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → Rel 𝐹)
Theorem	frn 6723	The range of a mapping. (Contributed by NM, 3-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
Theorem	frnd 6724	Deduction form of frn 6723. The range of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
Theorem	fdm 6725	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	fdmOLD 6726	Obsolete version of fdm 6725 as of 29-May-2024. (Contributed by NM, 2-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
Theorem	fdmd 6727	Deduction form of fdm 6725. The domain of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → dom 𝐹 = 𝐴)
Theorem	fdmi 6728	Inference associated with fdm 6725. The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
⊢ 𝐹:𝐴⟶𝐵    ⇒   ⊢ dom 𝐹 = 𝐴
Theorem	dffn3 6729	A function maps to its range. (Contributed by NM, 1-Sep-1999.)
⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
Theorem	ffrn 6730	A function maps to its range. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴⟶ran 𝐹)
Theorem	ffrnb 6731	Characterization of a function with domain and codomain (essentially using that the range is always included in the codomain). Generalization of ffrn 6730. (Contributed by BJ, 21-Sep-2024.)
⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹:𝐴⟶ran 𝐹 ∧ ran 𝐹 ⊆ 𝐵))
Theorem	ffrnbd 6732	A function maps to its range iff the range is a subset of its codomain. Generalization of ffrn 6730. (Contributed by AV, 20-Sep-2024.)
⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐴⟶ran 𝐹))
Theorem	fss 6733	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 6734	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 6735	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 6736	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 6737	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	fimacnv 6738	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 6739	Composition of a function with domain and codomain and a function as a function with domain and codomain. Generalization of fco 6740. (Contributed by AV, 18-Sep-2024.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐴)⟶𝐵)
Theorem	fco 6740	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	fcoOLD 6741	Obsolete version of fco 6740 as of 20-Sep-2024. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
Theorem	fcod 6742	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 6743	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 6744	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 6745	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 6746	A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))
Theorem	ffdmd 6747	The domain of a function. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → 𝐹:dom 𝐹⟶𝐵)
Theorem	fdmrn 6748	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 6749	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	fco3OLD 6750	Obsolete version of funcofd 6749 as 20-Sep-2024. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → Fun 𝐺)    ⇒   ⊢ (𝜑 → (𝐹 ∘ 𝐺):(◡𝐺 “ dom 𝐹)⟶ran 𝐹)
Theorem	opelf 6751	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 6752	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 6753	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 6754	The union of functions with disjoint domains is a function, deduction version of fun2 6753. (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 6755	Composition of two functions. (Contributed by NM, 22-May-2006.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)
Theorem	fssres 6756	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 6757	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 6758	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 6759	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 6760	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 6761	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 6762	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 6763	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 6764	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 6765	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 6766*	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 6767	The converse of a restriction of a function. (Contributed by NM, 26-Mar-1998.)
⊢ (𝐹:𝐴⟶𝐵 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ 𝐵))
Theorem	fimacnvdisj 6768	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 6769*	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 6770	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 6771	The empty function. (Contributed by NM, 14-Aug-1999.)
⊢ ∅:∅⟶𝐴
Theorem	f00 6772	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 6773	A function with empty domain is empty. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅)
Theorem	f0dom0 6774	A function is empty iff it has an empty domain. (Contributed by AV, 10-Feb-2019.)
⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅))
Theorem	f0rn0 6775*	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 6776	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 6777	A Cartesian product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
Theorem	fnconstg 6778	A Cartesian product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}) Fn 𝐴)
Theorem	fconst6g 6779	Constant function with loose range. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶𝐶)
Theorem	fconst6 6780	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 6781	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 6782	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))
Theorem	f1eq3 6783	Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)
⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵))
Theorem	nff1 6784	Bound-variable hypothesis builder for a one-to-one function. (Contributed by NM, 16-May-2004.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐹:𝐴–1-1→𝐵
Theorem	dff12 6785*	Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦))
Theorem	f1f 6786	A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵)
Theorem	f1fn 6787	A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴)
Theorem	f1fun 6788	A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹)
Theorem	f1rel 6789	A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
⊢ (𝐹:𝐴–1-1→𝐵 → Rel 𝐹)
Theorem	f1dm 6790	The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.) (Proof shortened by Wolf Lammen, 29-May-2024.)
⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴)
Theorem	f1dmOLD 6791	Obsolete version of f1dm 6790 as of 29-May-2024. (Contributed by NM, 8-Mar-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴)
Theorem	f1ss 6792	A function that is one-to-one is also one-to-one on some superset of its codomain. (Contributed by Mario Carneiro, 12-Jan-2013.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶)
Theorem	f1ssr 6793	A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Stefan O'Rear, 20-Feb-2015.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶)
Theorem	f1ssres 6794	A function that is one-to-one is also one-to-one on any subclass of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵)
Theorem	f1resf1 6795	The restriction of an injective function is injective. (Contributed by AV, 28-Jun-2022.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐷)
Theorem	f1cnvcnv 6796	Two ways to express that a set 𝐴 (not necessarily a function) is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one. We do not introduce a separate notation since we rarely use it. (Contributed by NM, 13-Aug-2004.)
⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴))
Theorem	f1cof1 6797	Composition of two one-to-one functions. Generalization of f1co 6798. (Contributed by AV, 18-Sep-2024.)
⊢ ((𝐹:𝐶–1-1→𝐷 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):(◡𝐺 “ 𝐶)–1-1→𝐷)
Theorem	f1co 6798	Composition of one-to-one functions when the codomain of the first matches the domain of the second. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.) (Proof shortened by AV, 20-Sep-2024.)
⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶)
Theorem	f1coOLD 6799	Obsolete version of f1co 6798 as of 20-Sep-2024. (Contributed by NM, 28-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶)
Theorem	foeq1 6800	Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹 = 𝐺 → (𝐹:𝐴–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵))