P. 53 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5201-5300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	funimass2 5201	A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ (◡𝐹 “ 𝐵)) → (𝐹 “ 𝐴) ⊆ 𝐵)
Theorem	imadiflem 5202	One direction of imadif 5203. This direction does not require Fun ◡𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)
⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∖ 𝐵))
Theorem	imadif 5203	The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.)
⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)))
Theorem	imainlem 5204	One direction of imain 5205. This direction does not require Fun ◡𝐹. (Contributed by Jim Kingdon, 25-Dec-2018.)
⊢ (𝐹 “ (𝐴 ∩ 𝐵)) ⊆ ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵))
Theorem	imain 5205	The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.)
⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) = ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵)))
Theorem	funimaexglem 5206	Lemma for funimaexg 5207. It constitutes the interesting part of funimaexg 5207, in which 𝐵 ⊆ dom 𝐴. (Contributed by Jim Kingdon, 27-Dec-2018.)
⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ dom 𝐴) → (𝐴 “ 𝐵) ∈ V)
Theorem	funimaexg 5207	Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 10-Sep-2006.)
⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)
Theorem	funimaex 5208	The image of a set under any function is also a set. Equivalent of Axiom of Replacement. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 17-Nov-2002.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)
Theorem	isarep1 5209*	Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by 𝜑(𝑥, 𝑦) i.e. the class ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑)
Theorem	isarep2 5210*	Part of a study of the Axiom of Replacement used by the Isabelle prover. In Isabelle, the sethood of PrimReplace is apparently postulated implicitly by its type signature "[ i, [ i, i ] => o ] => i", which automatically asserts that it is a set without using any axioms. To prove that it is a set in Metamath, we need the hypotheses of Isabelle's "Axiom of Replacement" as well as the Axiom of Replacement in the form funimaex 5208. (Contributed by NM, 26-Oct-2006.)
⊢ 𝐴 ∈ V    &   ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧)    ⇒   ⊢ ∃𝑤 𝑤 = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴)
Theorem	fneq1 5211	Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))
Theorem	fneq2 5212	Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵))
Theorem	fneq1d 5213	Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → 𝐹 = 𝐺)    ⇒   ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))
Theorem	fneq2d 5214	Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵))
Theorem	fneq12d 5215	Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐵))
Theorem	fneq12 5216	Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)
⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵) → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐵))
Theorem	fneq1i 5217	Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐹 = 𝐺    ⇒   ⊢ (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)
Theorem	fneq2i 5218	Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵)
Theorem	nffn 5219	Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝐹 Fn 𝐴
Theorem	fnfun 5220	A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
Theorem	fnrel 5221	A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
Theorem	fndm 5222	The domain of a function. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
Theorem	funfni 5223	Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004.)
⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → 𝜑)    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝜑)
Theorem	fndmu 5224	A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐵) → 𝐴 = 𝐵)
Theorem	fnbr 5225	The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ 𝐴)
Theorem	fnop 5226	The first argument of an ordered pair in a function belongs to the function's domain. (Contributed by NM, 8-Aug-1994.)
⊢ ((𝐹 Fn 𝐴 ∧ ⟨𝐵, 𝐶⟩ ∈ 𝐹) → 𝐵 ∈ 𝐴)
Theorem	fneu 5227*	There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑦 𝐵𝐹𝑦)
Theorem	fneu2 5228*	There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑦⟨𝐵, 𝑦⟩ ∈ 𝐹)
Theorem	fnun 5229	The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))
Theorem	fnunsn 5230	Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ (𝜑 → 𝑋 ∈ V)    &   ⊢ (𝜑 → 𝑌 ∈ V)    &   ⊢ (𝜑 → 𝐹 Fn 𝐷)    &   ⊢ 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})    &   ⊢ 𝐸 = (𝐷 ∪ {𝑋})    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷)    ⇒   ⊢ (𝜑 → 𝐺 Fn 𝐸)
Theorem	fnco 5231	Composition of two functions. (Contributed by NM, 22-May-2006.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)
Theorem	fnresdm 5232	A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
Theorem	fnresdisj 5233	A function restricted to a class disjoint with its domain is empty. (Contributed by NM, 23-Sep-2004.)
⊢ (𝐹 Fn 𝐴 → ((𝐴 ∩ 𝐵) = ∅ ↔ (𝐹 ↾ 𝐵) = ∅))
Theorem	2elresin 5234	Membership in two functions restricted by each other's domain. (Contributed by NM, 8-Aug-1994.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → ((⟨𝑥, 𝑦⟩ ∈ 𝐹 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐺) ↔ (⟨𝑥, 𝑦⟩ ∈ (𝐹 ↾ (𝐴 ∩ 𝐵)) ∧ ⟨𝑥, 𝑧⟩ ∈ (𝐺 ↾ (𝐴 ∩ 𝐵)))))
Theorem	fnssresb 5235	Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007.)
⊢ (𝐹 Fn 𝐴 → ((𝐹 ↾ 𝐵) Fn 𝐵 ↔ 𝐵 ⊆ 𝐴))
Theorem	fnssres 5236	Restriction of a function with a subclass of its domain. (Contributed by NM, 2-Aug-1994.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 ↾ 𝐵) Fn 𝐵)
Theorem	fnresin1 5237	Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴 ∩ 𝐵)) Fn (𝐴 ∩ 𝐵))
Theorem	fnresin2 5238	Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐵 ∩ 𝐴)) Fn (𝐵 ∩ 𝐴))
Theorem	fnres 5239*	An equivalence for functionality of a restriction. Compare dffun8 5151. (Contributed by Mario Carneiro, 20-May-2015.)
⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦)
Theorem	fnresi 5240	Functionality and domain of restricted identity. (Contributed by NM, 27-Aug-2004.)
⊢ ( I ↾ 𝐴) Fn 𝐴
Theorem	fnima 5241	The image of a function's domain is its range. (Contributed by NM, 4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (𝐹 Fn 𝐴 → (𝐹 “ 𝐴) = ran 𝐹)
Theorem	fn0 5242	A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (𝐹 Fn ∅ ↔ 𝐹 = ∅)
Theorem	fnimadisj 5243	A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)
⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 “ 𝐶) = ∅)
Theorem	fnimaeq0 5244	Images under a function never map nonempty sets to empty sets. (Contributed by Stefan O'Rear, 21-Jan-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝐹 “ 𝐵) = ∅ ↔ 𝐵 = ∅))
Theorem	dfmpt3 5245	Alternate definition for the maps-to notation df-mpt 3991. (Contributed by Mario Carneiro, 30-Dec-2016.)
⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵})
Theorem	fnopabg 5246*	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 5247*	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 5248*	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 5249*	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	mpt0 5250	A mapping operation with empty domain. (Contributed by Mario Carneiro, 28-Dec-2014.)
⊢ (𝑥 ∈ ∅ ↦ 𝐴) = ∅
Theorem	fnmpti 5251*	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 5252*	Domain of an ordered-pair class abstraction that specifies a function. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ dom 𝐹 = 𝐴
Theorem	dmmptd 5253*	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 5254	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	feq1 5255	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵))
Theorem	feq2 5256	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐴 = 𝐵 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶))
Theorem	feq3 5257	Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐴 = 𝐵 → (𝐹:𝐶⟶𝐴 ↔ 𝐹:𝐶⟶𝐵))
Theorem	feq23 5258	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 5259	Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
⊢ (𝜑 → 𝐹 = 𝐺)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵))
Theorem	feq2d 5260	Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶))
Theorem	feq3d 5261	Equality deduction for functions. (Contributed by AV, 1-Jan-2020.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹:𝑋⟶𝐴 ↔ 𝐹:𝑋⟶𝐵))
Theorem	feq12d 5262	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 5263	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 5264	Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐶⟶𝐷))
Theorem	feq1i 5265	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐹 = 𝐺    ⇒   ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵)
Theorem	feq2i 5266	Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐹:𝐴⟶𝐶 ↔ 𝐹:𝐵⟶𝐶)
Theorem	feq23i 5267	Equality inference for functions. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ (𝐹:𝐴⟶𝐵 ↔ 𝐹:𝐶⟶𝐷)
Theorem	feq23d 5268	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 5269	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 5270*	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 5271*	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	ffn 5272	A mapping is a function. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
Theorem	ffnd 5273	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 5274	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 5275	A mapping is a function. (Contributed by NM, 3-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
Theorem	ffund 5276	A mapping is a function, deduction version. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → Fun 𝐹)
Theorem	frel 5277	A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → Rel 𝐹)
Theorem	fdm 5278	The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
Theorem	fdmd 5279	Deduction form of fdm 5278. The domain of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → dom 𝐹 = 𝐴)
Theorem	fdmi 5280	The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
⊢ 𝐹:𝐴⟶𝐵    ⇒   ⊢ dom 𝐹 = 𝐴
Theorem	frn 5281	The range of a mapping. (Contributed by NM, 3-Aug-1994.)
⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
Theorem	frnd 5282	Deduction form of frn 5281. The range of a mapping. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → ran 𝐹 ⊆ 𝐵)
Theorem	dffn3 5283	A function maps to its range. (Contributed by NM, 1-Sep-1999.)
⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
Theorem	fss 5284	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 5285	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 5286	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 5287	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	fco 5288	Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ ((𝐹:𝐵⟶𝐶 ∧ 𝐺:𝐴⟶𝐵) → (𝐹 ∘ 𝐺):𝐴⟶𝐶)
Theorem	fco2 5289	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 5290	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	fex2 5291	A function with bounded domain and range is a set. This version is proven without the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)
Theorem	funssxp 5292	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 5293	A mapping is a partial function. (Contributed by NM, 25-Nov-2007.)
⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))
Theorem	opelf 5294	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 5295	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 5296	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	fnfco 5297	Composition of two functions. (Contributed by NM, 22-May-2006.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺:𝐵⟶𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)
Theorem	fssres 5298	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 5299	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 5300	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)