P. 69 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 6801-6900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	f1oun 6801	The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)
⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷))
Theorem	f1un 6802	The union of two one-to-one functions with disjoint domains and codomains. (Contributed by BTernaryTau, 3-Dec-2024.)
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐺:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1→(𝐵 ∪ 𝐷))
Theorem	resdif 6803	The restriction of a one-to-one onto function to a difference maps onto the difference of the images. (Contributed by Paul Chapman, 11-Apr-2009.)
⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐶 ∖ 𝐷))
Theorem	resin 6804	The restriction of a one-to-one onto function to an intersection maps onto the intersection of the images. (Contributed by Paul Chapman, 11-Apr-2009.)
⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∩ 𝐵)):(𝐴 ∩ 𝐵)–1-1-onto→(𝐶 ∩ 𝐷))
Theorem	f1oco 6805	Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.)
⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝐺:𝐴–1-1-onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1-onto→𝐶)
Theorem	f1cnv 6806	The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)
⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–1-1-onto→𝐴)
Theorem	funcocnv2 6807	Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
Theorem	fococnv2 6808	The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵))
Theorem	f1ococnv2 6809	The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by NM, 13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵))
Theorem	f1cocnv2 6810	Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
Theorem	f1ococnv1 6811	The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain. (Contributed by NM, 13-Dec-2003.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))
Theorem	f1cocnv1 6812	Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
⊢ (𝐹:𝐴–1-1→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))
Theorem	funcoeqres 6813	Express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ ((Fun 𝐺 ∧ (𝐹 ∘ 𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻 ∘ ◡𝐺))
Theorem	f1ssf1 6814	A subset of an injective function is injective. (Contributed by AV, 20-Nov-2020.)
⊢ ((Fun 𝐹 ∧ Fun ◡𝐹 ∧ 𝐺 ⊆ 𝐹) → Fun ◡𝐺)
Theorem	f10 6815	The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
⊢ ∅:∅–1-1→𝐴
Theorem	f10d 6816	The empty set maps one-to-one into any class, deduction version. (Contributed by AV, 25-Nov-2020.)
⊢ (𝜑 → 𝐹 = ∅)    ⇒   ⊢ (𝜑 → 𝐹:dom 𝐹–1-1→𝐴)
Theorem	f1o00 6817	One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Theorem	fo00 6818	Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
⊢ (𝐹:∅–onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Theorem	f1o0 6819	One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
⊢ ∅:∅–1-1-onto→∅
Theorem	f1oi 6820	A restriction of the identity relation is a one-to-one onto function. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) Avoid ax-12 2185. (Revised by TM, 10-Feb-2026.)
⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴
Theorem	f1oiOLD 6821	Obsolete version of f1oi 6820 as of 10-Feb-2026. (Contributed by NM, 30-Apr-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴
Theorem	f1ovi 6822	The identity relation is a one-to-one onto function on the universe. (Contributed by NM, 16-May-2004.)
⊢ I :V–1-1-onto→V
Theorem	f1osn 6823	A singleton of an ordered pair is one-to-one onto function. (Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵}
Theorem	f1osng 6824	A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})
Theorem	f1sng 6825	A singleton of an ordered pair is a one-to-one function. (Contributed by AV, 17-Apr-2021.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1→𝑊)
Theorem	fsnd 6826	A singleton of an ordered pair is a function. (Contributed by AV, 17-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → {⟨𝐴, 𝐵⟩}:{𝐴}⟶𝑊)
Theorem	f1oprswap 6827	A two-element swap is a bijection on a pair. (Contributed by Mario Carneiro, 23-Jan-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵})
Theorem	f1oprg 6828	An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 6827. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}:{𝐴, 𝐶}–1-1-onto→{𝐵, 𝐷}))
Theorem	tz6.12-2 6829*	Function value when 𝐹 is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2147, ax-11 2163, ax-12 2185. (Revised by TM, 25-Jan-2026.)
⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)
Theorem	tz6.12-2OLD 6830*	Obsolete version of tz6.12-2 6829 as of 25-Jan-2026. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)
Theorem	fveu 6831*	The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
Theorem	brprcneu 6832*	If 𝐴 is a proper class and 𝐹 is any class, then there is no unique set which is related to 𝐴 through the binary relation 𝐹. See brprcneuALT 6833 for a proof that uses ax-pow 5312 instead of ax-pr 5379. (Contributed by Scott Fenton, 7-Oct-2017.)
⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥)
Theorem	brprcneuALT 6833*	Alternate proof of brprcneu 6832 using ax-pow 5312 instead of ax-pr 5379. (Contributed by Scott Fenton, 7-Oct-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥)
Theorem	fvprc 6834	A function's value at a proper class is the empty set. See fvprcALT 6835 for a proof that uses ax-pow 5312 instead of ax-pr 5379. (Contributed by NM, 20-May-1998.) Avoid ax-pow 5312. (Revised by BTernaryTau, 3-Aug-2024.) (Proof shortened by BTernaryTau, 3-Dec-2024.)
⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
Theorem	fvprcALT 6835	Alternate proof of fvprc 6834 using ax-pow 5312 instead of ax-pr 5379. (Contributed by NM, 20-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
Theorem	rnfvprc 6836	The range of a function value at a proper class is empty. (Contributed by AV, 20-Aug-2022.)
⊢ 𝑌 = (𝐹‘𝑋)    ⇒   ⊢ (¬ 𝑋 ∈ V → ran 𝑌 = ∅)
Theorem	fv2 6837*	Alternate definition of function value. Definition 10.11 of [Quine] p. 68. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥)}
Theorem	dffv3 6838*	A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.)
⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))
Theorem	dffv4 6839*	The previous definition of function value, from before the ℩ operator was introduced. Although based on the idea embodied by Definition 10.2 of [Quine] p. 65 (see args 6059), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
Theorem	elfv 6840*	Membership in a function value. (Contributed by NM, 30-Apr-2004.)
⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)))
Theorem	fveq1 6841	Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
⊢ (𝐹 = 𝐺 → (𝐹‘𝐴) = (𝐺‘𝐴))
Theorem	fveq2 6842	Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
⊢ (𝐴 = 𝐵 → (𝐹‘𝐴) = (𝐹‘𝐵))
Theorem	fveq1i 6843	Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
⊢ 𝐹 = 𝐺    ⇒   ⊢ (𝐹‘𝐴) = (𝐺‘𝐴)
Theorem	fveq1d 6844	Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
⊢ (𝜑 → 𝐹 = 𝐺)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))
Theorem	fveq2i 6845	Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐹‘𝐴) = (𝐹‘𝐵)
Theorem	fveq2d 6846	Equality deduction for function value. (Contributed by NM, 29-May-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵))
Theorem	2fveq3 6847	Equality theorem for nested function values. (Contributed by AV, 14-Aug-2022.)
⊢ (𝐴 = 𝐵 → (𝐹‘(𝐺‘𝐴)) = (𝐹‘(𝐺‘𝐵)))
Theorem	fveq12i 6848	Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
⊢ 𝐹 = 𝐺    &   ⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐹‘𝐴) = (𝐺‘𝐵)
Theorem	fveq12d 6849	Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐵))
Theorem	fveqeq2d 6850	Equality deduction for function value. (Contributed by BJ, 30-Aug-2022.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐴) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶))
Theorem	fveqeq2 6851	Equality deduction for function value. (Contributed by BJ, 31-Aug-2022.)
⊢ (𝐴 = 𝐵 → ((𝐹‘𝐴) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶))
Theorem	nffv 6852	Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥(𝐹‘𝐴)
Theorem	nffvmpt1 6853*	Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐶)
Theorem	nffvd 6854	Deduction version of bound-variable hypothesis builder nffv 6852. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐹)    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝐹‘𝐴))
Theorem	fvex 6855	The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by NM, 30-Dec-1996.)
⊢ (𝐹‘𝐴) ∈ V
Theorem	fvexi 6856	The value of a class exists. Inference form of fvex 6855. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ 𝐴 = (𝐹‘𝐵)    ⇒   ⊢ 𝐴 ∈ V
Theorem	fvexd 6857	The value of a class exists (as consequent of anything). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → (𝐹‘𝐴) ∈ V)
Theorem	fvif 6858	Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵))
Theorem	iffv 6859	Move a conditional outside of a function. (Contributed by Thierry Arnoux, 28-Sep-2018.)
⊢ (if(𝜑, 𝐹, 𝐺)‘𝐴) = if(𝜑, (𝐹‘𝐴), (𝐺‘𝐴))
Theorem	fv3 6860*	Alternate definition of the value of a function. Definition 6.11 of [TakeutiZaring] p. 26. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐹‘𝐴) = {𝑥 ∣ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)}
Theorem	fvres 6861	The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
Theorem	fvresd 6862	The value of a restricted function, deduction version of fvres 6861. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
Theorem	funssfv 6863	The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)
⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴))
Theorem	tz6.12c 6864*	Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by SN, 23-Dec-2024.)
⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
Theorem	tz6.12-1 6865*	Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by SN, 23-Dec-2024.)
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦)
Theorem	tz6.12 6866*	Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)
⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑦)
Theorem	tz6.12f 6867*	Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)
⊢ Ⅎ𝑦𝐹    ⇒   ⊢ ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹‘𝐴) = 𝑦)
Theorem	tz6.12i 6868	Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ (𝐵 ≠ ∅ → ((𝐹‘𝐴) = 𝐵 → 𝐴𝐹𝐵))
Theorem	fvbr0 6869	Two possibilities for the behavior of a function value. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
⊢ (𝑋𝐹(𝐹‘𝑋) ∨ (𝐹‘𝑋) = ∅)
Theorem	fvrn0 6870	A function value is a member of the range plus null. (Contributed by Scott Fenton, 8-Jun-2011.) (Revised by Stefan O'Rear, 3-Jan-2015.)
⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})
Theorem	fvn0fvelrn 6871	If the value of a function is not null, the value is an element of the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Proof shortened by SN, 13-Jan-2025.)
⊢ ((𝐹‘𝑋) ≠ ∅ → (𝐹‘𝑋) ∈ ran 𝐹)
Theorem	elfvunirn 6872	A function value is a subset of the union of the range. (An artifact of our function value definition, compare elfvdm 6876). (Contributed by Thierry Arnoux, 13-Nov-2016.) Remove functionhood antecedent. (Revised by SN, 10-Jan-2025.)
⊢ (𝐵 ∈ (𝐹‘𝐴) → 𝐵 ∈ ∪ ran 𝐹)
Theorem	fvssunirn 6873	The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 31-Aug-2015.) (Proof shortened by SN, 13-Jan-2025.)
⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹
Theorem	ndmfv 6874	The value of a class outside its domain is the empty set. (An artifact of our function value definition.) (Contributed by NM, 24-Aug-1995.)
⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
Theorem	ndmfvrcl 6875	Reverse closure law for function with the empty set not in its domain (if 𝑅 = 𝑆). (Contributed by NM, 26-Apr-1996.) The class containing the function value does not have to be the domain. (Revised by Zhi Wang, 10-Nov-2025.)
⊢ dom 𝐹 = 𝑆    &   ⊢ ¬ ∅ ∈ 𝑅    ⇒   ⊢ ((𝐹‘𝐴) ∈ 𝑅 → 𝐴 ∈ 𝑆)
Theorem	elfvdm 6876	If a function value has a member, then the argument belongs to the domain. (An artifact of our function value definition.) (Contributed by NM, 12-Feb-2007.) (Proof shortened by BJ, 22-Oct-2022.)
⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ dom 𝐹)
Theorem	elfvex 6877	If a function value has a member, then the argument is a set. (An artifact of our function value definition.) (Contributed by Mario Carneiro, 6-Nov-2015.)
⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐵 ∈ V)
Theorem	elfvexd 6878	If a function value has a member, then its argument is a set. Deduction form of elfvex 6877. (An artifact of our function value definition.) (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (𝐵‘𝐶))    ⇒   ⊢ (𝜑 → 𝐶 ∈ V)
Theorem	eliman0 6879	A nonempty function value is an element of the image of the function. (Contributed by Thierry Arnoux, 25-Jun-2019.)
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ (𝐹‘𝐴) = ∅) → (𝐹‘𝐴) ∈ (𝐹 “ 𝐵))
Theorem	nfvres 6880	The value of a non-member of a restriction is the empty set. (An artifact of our function value definition.) (Contributed by NM, 13-Nov-1995.)
⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅)
Theorem	nfunsn 6881	If the restriction of a class to a singleton is not a function, then its value is the empty set. (An artifact of our function value definition.) (Contributed by NM, 8-Aug-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅)
Theorem	fvfundmfvn0 6882	If the "value of a class" at an argument is not the empty set, then the argument is in the domain of the class and the class restricted to the singleton formed on that argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.) (Proof shortened by BJ, 13-Aug-2022.)
⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
Theorem	0fv 6883	Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
⊢ (∅‘𝐴) = ∅
Theorem	fv2prc 6884	A function value of a function value at a proper class is the empty set. (Contributed by AV, 8-Apr-2021.)
⊢ (¬ 𝐴 ∈ V → ((𝐹‘𝐴)‘𝐵) = ∅)
Theorem	elfv2ex 6885	If a function value of a function value has a member, then the first argument is a set. (Contributed by AV, 8-Apr-2021.)
⊢ (𝐴 ∈ ((𝐹‘𝐵)‘𝐶) → 𝐵 ∈ V)
Theorem	fveqres 6886	Equal values imply equal values in a restriction. (Contributed by NM, 13-Nov-1995.)
⊢ ((𝐹‘𝐴) = (𝐺‘𝐴) → ((𝐹 ↾ 𝐵)‘𝐴) = ((𝐺 ↾ 𝐵)‘𝐴))
Theorem	csbfv12 6887	Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.) (Revised by NM, 20-Aug-2018.)
⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)
Theorem	csbfv2g 6888*	Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (𝐹‘⦋𝐴 / 𝑥⦌𝐵))
Theorem	csbfv 6889*	Substitution for a function value. (Contributed by NM, 1-Jan-2006.) (Revised by NM, 20-Aug-2018.)
⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴)
Theorem	funbrfv 6890	The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵))
Theorem	funopfv 6891	The second element in an ordered pair member of a function is the function's value. (Contributed by NM, 19-Jul-1996.)
⊢ (Fun 𝐹 → (⟨𝐴, 𝐵⟩ ∈ 𝐹 → (𝐹‘𝐴) = 𝐵))
Theorem	fnbrfvb 6892	Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ 𝐵𝐹𝐶))
Theorem	fnopfvb 6893	Equivalence of function value and ordered pair membership. (Contributed by NM, 7-Nov-1995.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐹))
Theorem	fvelima2 6894*	Function value in an image. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ (𝐹 “ 𝐶)) → ∃𝑥 ∈ (𝐴 ∩ 𝐶)(𝐹‘𝑥) = 𝐵)
Theorem	funbrfvb 6895	Equivalence of function value and binary relation. (Contributed by NM, 26-Mar-2006.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) = 𝐵 ↔ 𝐴𝐹𝐵))
Theorem	funopfvb 6896	Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by NM, 26-Jan-1997.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) = 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝐹))
Theorem	fnbrfvb2 6897	Version of fnbrfvb 6892 for functions on Cartesian products: function value expressed as a binary relation. See fnbrovb 7419 for the form when 𝐹 is seen as a binary operation. (Contributed by BJ, 15-Feb-2022.)
⊢ ((𝐹 Fn (𝑉 × 𝑊) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → ((𝐹‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩𝐹𝐶))
Theorem	fdmeu 6898*	There is exactly one codomain element for each element of the domain of a function. (Contributed by AV, 20-Apr-2025.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 (𝐹‘𝑋) = 𝑦)
Theorem	funbrfv2b 6899	Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹‘𝐴) = 𝐵)))
Theorem	dffn5 6900*	Representation of a function in terms of its values. (Contributed by FL, 14-Sep-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))