P. 66 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 6501-6600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dffun6 6501*	Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) Avoid ax-10 2147, ax-12 2185. (Revised by SN, 19-Dec-2024.)
⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
Theorem	dffun3 6502*	Alternate definition of function. (Contributed by NM, 29-Dec-1996.) (Proof shortened by SN, 19-Dec-2024.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧)))
Theorem	dffun4 6503*	Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧)))
Theorem	dffun5 6504*	Alternate definition of function. (Contributed by NM, 29-Dec-1996.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧)))
Theorem	dffun6f 6505*	Definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    ⇒   ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦))
Theorem	funmo 6506*	A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) (Proof shortened by SN, 19-Dec-2024.)
⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦)
Theorem	funrel 6507	A function is a relation. (Contributed by NM, 1-Aug-1994.)
⊢ (Fun 𝐴 → Rel 𝐴)
Theorem	0nelfun 6508	A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.)
⊢ (Fun 𝑅 → ∅ ∉ 𝑅)
Theorem	funss 6509	Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)
⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴))
Theorem	funeq 6510	Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))
Theorem	funeqi 6511	Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (Fun 𝐴 ↔ Fun 𝐵)
Theorem	funeqd 6512	Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (Fun 𝐴 ↔ Fun 𝐵))
Theorem	nffun 6513	Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.)
⊢ Ⅎ𝑥𝐹    ⇒   ⊢ Ⅎ𝑥Fun 𝐹
Theorem	sbcfung 6514	Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝐴 / 𝑥⦌𝐹))
Theorem	funeu 6515*	There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦)
Theorem	funeu2 6516*	There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.)
⊢ ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹) → ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹)
Theorem	dffun7 6517*	Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one". However, dffun8 6518 shows that it does not matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦))
Theorem	dffun8 6518*	Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 6517. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦))
Theorem	dffun9 6519*	Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.)
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦))
Theorem	funfn 6520	A class is a function if and only if it is a function on its domain. (Contributed by NM, 13-Aug-2004.)
⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴)
Theorem	funfnd 6521	A function is a function on its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → Fun 𝐴)    ⇒   ⊢ (𝜑 → 𝐴 Fn dom 𝐴)
Theorem	funi 6522	The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6618. (Contributed by NM, 30-Apr-1998.)
⊢ Fun I
Theorem	nfunv 6523	The universal class is not a function. (Contributed by Raph Levien, 27-Jan-2004.)
⊢ ¬ Fun V
Theorem	funopg 6524	A Kuratowski ordered pair of sets is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 6541, as relsnopg 5750 is to relop 5797. (New usage is discouraged.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ Fun ⟨𝐴, 𝐵⟩) → 𝐴 = 𝐵)
Theorem	funopab 6525*	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 6526*	A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
⊢ Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
Theorem	funopab4 6527*	A class of ordered pairs of values in the form used by df-mpt 5168 is a function. (Contributed by NM, 17-Feb-2013.)
⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)}
Theorem	funmpt 6528	A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.)
⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵)
Theorem	funmpt2 6529	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 6530	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 6531	Composition of two functions, generalization of funco 6530. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
⊢ ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺))
Theorem	funres 6532	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 6533	A restriction of a function is a function. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ (𝜑 → Fun 𝐹)    ⇒   ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴))
Theorem	funssres 6534	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 6535	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 6536	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 6537*	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 6538	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 6539	A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.)
⊢ (Fun 𝐹 → Fun (𝐹 ∖ 𝐴))
Theorem	funcnvsn 6540	The converse singleton of an ordered pair is a function. This is equivalent to funsn 6543 via cnvsn 6182, 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 6541	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 6542	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 6543	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 6544	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 6545	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 6546	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 6547	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 6548	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 6549	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 6550	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 6551	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 6552	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 6553	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 6554	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 6555	The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.)
⊢ Fun ∅
Theorem	funcnv0 6556	The converse of the empty set is a function. (Contributed by AV, 7-Jan-2021.)
⊢ Fun ◡∅
Theorem	funcnvcnv 6557	The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.)
⊢ (Fun 𝐴 → Fun ◡◡𝐴)
Theorem	funcnv2 6558*	A simpler equivalence for single-rooted (see funcnv 6559). (Contributed by NM, 9-Aug-2004.)
⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦)
Theorem	funcnv 6559*	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 6558 for a simpler version. (Contributed by NM, 13-Aug-2004.)
⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦)
Theorem	funcnv3 6560*	A condition showing a class is single-rooted. (See funcnv 6559). (Contributed by NM, 26-May-2006.)
⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦)
Theorem	fun2cnv 6561*	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 6562	A single-valued relation is a function. (See fun2cnv 6561 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 6563*	Single-rootedness (see funcnv 6559) 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 6564*	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 6565*	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 6566	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 6567	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 6568	The converse of a restricted function. (Contributed by NM, 27-Mar-1998.)
⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ (𝐹 “ 𝐴)))
Theorem	cnvresid 6569	Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.)
⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴)
Theorem	funcnvres2 6570	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 6571	The image of the preimage of a function. (Contributed by NM, 25-May-2004.)
⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹))
Theorem	funimass1 6572	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 6573	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 6574	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 6575	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 6576	Condition for the range of a one-to-one function to be the range of one its restrictions. Variant of imadifssran 6107. (Contributed by AV, 4-Oct-2025.)
⊢ (Fun ◡𝐹 → ((𝐹 “ (dom 𝐹 ∖ 𝐴)) ⊆ ran (𝐹 ↾ 𝐴) → ran 𝐹 = ran (𝐹 ↾ 𝐴)))
Theorem	funimaexg 6577	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 2147, ax-12 2185. (Revised by SN, 19-Dec-2024.)
⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V)
Theorem	funimaex 6578	The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5212. 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 6579*	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	isarep2 6580*	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 6578. (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 6581	Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))
Theorem	fneq2 6582	Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵))
Theorem	fneq1d 6583	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 6584	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 6585	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 6586	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 6587	Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
⊢ 𝐹 = 𝐺    ⇒   ⊢ (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)
Theorem	fneq2i 6588	Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵)
Theorem	nffn 6589	Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝐹 Fn 𝐴
Theorem	fnfun 6590	A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
Theorem	fnfund 6591	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 6592	A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
Theorem	fndm 6593	The domain of a function. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
Theorem	fndmi 6594	The domain of a function. (Contributed by Wolf Lammen, 1-Jun-2024.)
⊢ 𝐹 Fn 𝐴    ⇒   ⊢ dom 𝐹 = 𝐴
Theorem	fndmd 6595	The domain of a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    ⇒   ⊢ (𝜑 → dom 𝐹 = 𝐴)
Theorem	funfni 6596	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 6597	A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐵) → 𝐴 = 𝐵)
Theorem	fnbr 6598	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 6599	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 6600*	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)