P. 67 of Theorem List - Metamath Proof Explorer

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

Type	Label	Description
Statement
Theorem	funopab 6601*	A class of ordered pairs is a function when there is at most one second member for each pair. (Contributed by NM, 16-May-1995.)
⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∀𝑥∃*𝑦𝜑)
Theorem	funopabeq 6602*	A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
⊢ Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
Theorem	funopab4 6603*	A class of ordered pairs of values in the form used by df-mpt 5226 is a function. (Contributed by NM, 17-Feb-2013.)
⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)}
Theorem	funmpt 6604	A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	funmpt2 6605	Functionality of a class given by a maps-to notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ Fun 𝐹
Theorem	funco 6606	The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))
Theorem	funresfunco 6607	Composition of two functions, generalization of funco 6606. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
⊢ ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))
Theorem	funres 6608	A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.)
⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴))
Theorem	funresd 6609	A restriction of a function is a function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → Fun 𝐹)    ⇒   ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴))
Theorem	funssres 6610	The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.)
⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
Theorem	fun2ssres 6611	Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)
⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
Theorem	funun 6612	The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.)
⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))
Theorem	fununmo 6613*	If the union of classes is a function, there is at most one element in relation to an arbitrary element regarding one of these classes. (Contributed by AV, 18-Jul-2019.)
⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐹𝑦)
Theorem	fununfun 6614	If the union of classes is a function, the classes itselves are functions. (Contributed by AV, 18-Jul-2019.)
⊢ (Fun (𝐹 ∪ 𝐺) → (Fun 𝐹 ∧ Fun 𝐺))
Theorem	fundif 6615	A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.)
⊢ (Fun 𝐹 → Fun (𝐹 ∖ 𝐴))
Theorem	funcnvsn 6616	The converse singleton of an ordered pair is a function. This is equivalent to funsn 6619 via cnvsn 6246, but stating it this way allows to skip the sethood assumptions on 𝐴 and 𝐵. (Contributed by NM, 30-Apr-2015.)
⊢ Fun ◡{⟨𝐴, 𝐵⟩}
Theorem	funsng 6617	A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {⟨𝐴, 𝐵⟩})
Theorem	fnsng 6618	Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴})
Theorem	funsn 6619	A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ Fun {⟨𝐴, 𝐵⟩}
Theorem	funprg 6620	A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
Theorem	funtpg 6621	A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.) (Proof shortened by JJ, 14-Jul-2021.)
⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → Fun {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩})
Theorem	funpr 6622	A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    ⇒   ⊢ (𝐴 ≠ 𝐵 → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩})
Theorem	funtp 6623	A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    &   ⊢ 𝐸 ∈ V    &   ⊢ 𝐹 ∈ V    ⇒   ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → Fun {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})
Theorem	fnsn 6624	Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ {⟨𝐴, 𝐵⟩} Fn {𝐴}
Theorem	fnprg 6625	Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} Fn {𝐴, 𝐵})
Theorem	fntpg 6626	Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.)
⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} Fn {𝑋, 𝑌, 𝑍})
Theorem	fntp 6627	A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 ∈ V    &   ⊢ 𝐸 ∈ V    &   ⊢ 𝐹 ∈ V    ⇒   ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} Fn {𝐴, 𝐵, 𝐶})
Theorem	funcnvpr 6628	The converse pair of ordered pairs is a function if the second members are different. Note that the second members need not be sets. (Contributed by AV, 23-Jan-2021.)
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
Theorem	funcnvtp 6629	The converse triple of ordered pairs is a function if the second members are pairwise different. Note that the second members need not be sets. (Contributed by AV, 23-Jan-2021.)
⊢ (((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐷 ≠ 𝐹)) → Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩})
Theorem	funcnvqp 6630	The converse quadruple of ordered pairs is a function if the second members are pairwise different. Note that the second members need not be sets. (Contributed by AV, 23-Jan-2021.) (Proof shortened by JJ, 14-Jul-2021.)
⊢ ((((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉) ∧ (𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐻) ∧ (𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐻) ∧ 𝐹 ≠ 𝐻)) → Fun ◡({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩}))
Theorem	fun0 6631	The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
⊢ Fun ∅
Theorem	funcnv0 6632	The converse of the empty set is a function. (Contributed by AV, 7-Jan-2021.)
⊢ Fun ◡∅
Theorem	funcnvcnv 6633	The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
⊢ (Fun 𝐴 → Fun ◡◡𝐴)
Theorem	funcnv2 6634*	A simpler equivalence for single-rooted (see funcnv 6635). (Contributed by NM, 9-Aug-2004.)
⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
Theorem	funcnv 6635*	The converse of a class is a function iff the class is single-rooted, which means that for any 𝑦 in the range of 𝐴 there is at most one 𝑥 such that 𝑥𝐴𝑦. Definition of single-rooted in [Enderton] p. 43. See funcnv2 6634 for a simpler version. (Contributed by NM, 13-Aug-2004.)
⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)
Theorem	funcnv3 6636*	A condition showing a class is single-rooted. (See funcnv 6635). (Contributed by NM, 26-May-2006.)
⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦)
Theorem	fun2cnv 6637*	The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that 𝐴 is not necessarily a function. (Contributed by NM, 13-Aug-2004.)
⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦)
Theorem	svrelfun 6638	A single-valued relation is a function. (See fun2cnv 6637 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ Fun ◡◡𝐴))
Theorem	fncnv 6639*	Single-rootedness (see funcnv 6635) of a class cut down by a Cartesian product. (Contributed by NM, 5-Mar-2007.)
⊢ (◡(𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦)
Theorem	fun11 6640*	Two ways of stating that 𝐴 is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one (but not necessarily a function). (Contributed by NM, 17-Jan-2006.)
⊢ ((Fun ◡◡𝐴 ∧ Fun ◡𝐴) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑥 = 𝑧 ↔ 𝑦 = 𝑤)))
Theorem	fununi 6641*	The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.)
⊢ (∀𝑓 ∈ 𝐴 (Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ∪ 𝐴)
Theorem	funin 6642	The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (Fun 𝐹 → Fun (𝐹 ∩ 𝐺))
Theorem	funres11 6643	The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.)
⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐴))
Theorem	funcnvres 6644	The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ (𝐹 “ 𝐴)))
Theorem	cnvresid 6645	Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
Theorem	funcnvres2 6646	The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by NM, 4-May-2005.)
⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (𝐹 ↾ (◡𝐹 “ 𝐴)))
Theorem	funimacnv 6647	The image of the preimage of a function. (Contributed by NM, 25-May-2004.)
⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹))
Theorem	funimass1 6648	A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → ((◡𝐹 “ 𝐴) ⊆ 𝐵 → 𝐴 ⊆ (𝐹 “ 𝐵)))
Theorem	funimass2 6649	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	imadif 6650	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	imain 6651	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	f1imadifssran 6652	Condition for the range of a one-to-one function to be the range of one its restrictions. Variant of imadifssran 6171. (Contributed by AV, 4-Oct-2025.)
⊢ (Fun ◡𝐹 → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴)))
Theorem	funimaexg 6653	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.) Shorten proof and avoid ax-10 2141, ax-12 2177. (Revised by SN, 19-Dec-2024.)
⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)
Theorem	funimaexgOLD 6654	Obsolete version of funimaexg 6653 as of 19-Dec-2024. (Contributed by NM, 10-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)
Theorem	funimaex 6655	The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5279. 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 6656*	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.) (Proof shortened by SN, 19-Dec-2024.)
⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑)
Theorem	isarep1OLD 6657*	Obsolete version of isarep1 6656 as of 19-Dec-2024. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑)
Theorem	isarep2 6658*	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 6655. (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 6659	Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))
Theorem	fneq2 6660	Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵))
Theorem	fneq1d 6661	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 6662	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 6663	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 6664	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 6665	Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐹 = 𝐺    ⇒   ⊢ (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)
Theorem	fneq2i 6666	Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵)
Theorem	nffn 6667	Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝐹 Fn 𝐴
Theorem	fnfun 6668	A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
Theorem	fnfund 6669	A function with domain is a function, deduction form. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    ⇒   ⊢ (𝜑 → Fun 𝐹)
Theorem	fnrel 6670	A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
Theorem	fndm 6671	The domain of a function. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
Theorem	fndmi 6672	The domain of a function. (Contributed by Wolf Lammen, 1-Jun-2024.)
⊢ 𝐹 Fn 𝐴    ⇒   ⊢ dom 𝐹 = 𝐴
Theorem	fndmd 6673	The domain of a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    ⇒   ⊢ (𝜑 → dom 𝐹 = 𝐴)
Theorem	funfni 6674	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 6675	A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐵) → 𝐴 = 𝐵)
Theorem	fnbr 6676	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 6677	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 6678*	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 6679*	There is exactly one value of a function. (Contributed by NM, 7-Nov-1995.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃!𝑦⟨𝐵, 𝑦⟩ ∈ 𝐹)
Theorem	fnunres1 6680	Restriction of a disjoint union to the domain of the first function. (Contributed by Thierry Arnoux, 9-Dec-2021.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹)
Theorem	fnunres2 6681	Restriction of a disjoint union to the domain of the second function. (Contributed by Thierry Arnoux, 12-Oct-2023.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)
Theorem	fnun 6682	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	fnund 6683	The union of two functions with disjoint domains, a deduction version. (Contributed by metakunt, 28-May-2024.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    ⇒   ⊢ (𝜑 → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))
Theorem	fnunop 6684	Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.) (Revised by AV, 16-Aug-2024.)
⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 Fn 𝐷)    &   ⊢ 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})    &   ⊢ 𝐸 = (𝐷 ∪ {𝑋})    &   ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷)    ⇒   ⊢ (𝜑 → 𝐺 Fn 𝐸)
Theorem	fncofn 6685	Composition of a function with domain and a function as a function with domain. Generalization of fnco 6686. (Contributed by AV, 17-Sep-2024.)
⊢ ((𝐹 Fn 𝐴 ∧ Fun 𝐺) → (𝐹 ∘ 𝐺) Fn (◡𝐺 “ 𝐴))
Theorem	fnco 6686	Composition of two functions with domains as a function with domain. (Contributed by NM, 22-May-2006.) (Proof shortened by AV, 20-Sep-2024.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ran 𝐺 ⊆ 𝐴) → (𝐹 ∘ 𝐺) Fn 𝐵)
Theorem	fnresdm 6687	A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)
⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)
Theorem	fnresdisj 6688	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 6689	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 6690	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 6691	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	fnssresd 6692	Restriction of a function to a subclass of its domain. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐴)    ⇒   ⊢ (𝜑 → (𝐹 ↾ 𝐵) Fn 𝐵)
Theorem	fnresin1 6693	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 6694	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 6695*	An equivalence for functionality of a restriction. Compare dffun8 6594. (Contributed by Mario Carneiro, 20-May-2015.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦)
Theorem	idfn 6696	The identity relation is a function on the universal class. See also funi 6598. (Contributed by BJ, 23-Dec-2023.)
⊢ I Fn V
Theorem	fnresi 6697	The restricted identity relation is a function on the restricting class. (Contributed by NM, 27-Aug-2004.) (Proof shortened by BJ, 27-Dec-2023.)
⊢ ( I ↾ 𝐴) Fn 𝐴
Theorem	fnima 6698	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 6699	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 6700	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) = ∅)