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	f1ofn 6701	A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
Theorem	f1ofun 6702	A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹)
Theorem	f1orel 6703	A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹)
Theorem	f1odm 6704	The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴)
Theorem	dff1o2 6705	Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))
Theorem	dff1o3 6706	Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹))
Theorem	f1ofo 6707	A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵)
Theorem	dff1o4 6708	Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
Theorem	dff1o5 6709	Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵))
Theorem	f1orn 6710	A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)
⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹))
Theorem	f1f1orn 6711	A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
Theorem	f1ocnv 6712	The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
Theorem	f1ocnvb 6713	A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by NM, 8-Dec-2003.)
⊢ (Rel 𝐹 → (𝐹:𝐴–1-1-onto→𝐵 ↔ ◡𝐹:𝐵–1-1-onto→𝐴))
Theorem	f1ores 6714	The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶))
Theorem	f1orescnv 6715	The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)
⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → (◡𝐹 ↾ 𝑃):𝑃–1-1-onto→𝑅)
Theorem	f1imacnv 6716	Preimage of an image. (Contributed by NM, 30-Sep-2004.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝐶)) = 𝐶)
Theorem	foimacnv 6717	A reverse version of f1imacnv 6716. (Contributed by Jeff Hankins, 16-Jul-2009.)
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝐶)) = 𝐶)
Theorem	foun 6718	The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–onto→(𝐵 ∪ 𝐷))
Theorem	f1oun 6719	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	resdif 6720	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 6721	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 6722	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 6723	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 6724	Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
Theorem	fococnv2 6725	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 6726	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 6727	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 6728	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 6729	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 6730	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 6731	A subset of an injective function is injective. (Contributed by AV, 20-Nov-2020.)
⊢ ((Fun 𝐹 ∧ Fun ◡𝐹 ∧ 𝐺 ⊆ 𝐹) → Fun ◡𝐺)
Theorem	f10 6732	The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
⊢ ∅:∅–1-1→𝐴
Theorem	f10d 6733	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 6734	One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Theorem	fo00 6735	Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
⊢ (𝐹:∅–onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Theorem	f1o0 6736	One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
⊢ ∅:∅–1-1-onto→∅
Theorem	f1oi 6737	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.)
⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴
Theorem	f1ovi 6738	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 6739	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 6740	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 6741	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 6742	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 6743	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 6744	An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 6743. (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 6745*	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.)
⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)
Theorem	fveu 6746*	The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
Theorem	brprcneu 6747*	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. (Contributed by Scott Fenton, 7-Oct-2017.)
⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥)
Theorem	fvprc 6748	A function's value at a proper class is the empty set. See fvprcALT 6749 for a proof that uses ax-pow 5283 instead of ax-sep 5218 and ax-pr 5347. (Contributed by NM, 20-May-1998.) Avoid ax-pow 5283. (Revised by BTernaryTau, 3-Aug-2024.)
⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
Theorem	fvprcALT 6749	Alternate proof of fvprc 6748 using ax-pow 5283 instead of ax-sep 5218 and ax-pr 5347. (Contributed by NM, 20-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
Theorem	rnfvprc 6750	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 6751*	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 6752*	A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.)
⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))
Theorem	dffv4 6753*	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 5989), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}
Theorem	elfv 6754*	Membership in a function value. (Contributed by NM, 30-Apr-2004.)
⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)))
Theorem	fveq1 6755	Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
⊢ (𝐹 = 𝐺 → (𝐹‘𝐴) = (𝐺‘𝐴))
Theorem	fveq2 6756	Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
⊢ (𝐴 = 𝐵 → (𝐹‘𝐴) = (𝐹‘𝐵))
Theorem	fveq1i 6757	Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
⊢ 𝐹 = 𝐺    ⇒   ⊢ (𝐹‘𝐴) = (𝐺‘𝐴)
Theorem	fveq1d 6758	Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
⊢ (𝜑 → 𝐹 = 𝐺)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))
Theorem	fveq2i 6759	Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐹‘𝐴) = (𝐹‘𝐵)
Theorem	fveq2d 6760	Equality deduction for function value. (Contributed by NM, 29-May-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵))
Theorem	2fveq3 6761	Equality theorem for nested function values. (Contributed by AV, 14-Aug-2022.)
⊢ (𝐴 = 𝐵 → (𝐹‘(𝐺‘𝐴)) = (𝐹‘(𝐺‘𝐵)))
Theorem	fveq12i 6762	Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
⊢ 𝐹 = 𝐺    &   ⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐹‘𝐴) = (𝐺‘𝐵)
Theorem	fveq12d 6763	Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐵))
Theorem	fveqeq2d 6764	Equality deduction for function value. (Contributed by BJ, 30-Aug-2022.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐴) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶))
Theorem	fveqeq2 6765	Equality deduction for function value. (Contributed by BJ, 31-Aug-2022.)
⊢ (𝐴 = 𝐵 → ((𝐹‘𝐴) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶))
Theorem	nffv 6766	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 6767*	Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐶)
Theorem	nffvd 6768	Deduction version of bound-variable hypothesis builder nffv 6766. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐹)    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝐹‘𝐴))
Theorem	fvex 6769	The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by NM, 30-Dec-1996.)
⊢ (𝐹‘𝐴) ∈ V
Theorem	fvexi 6770	The value of a class exists. Inference form of fvex 6769. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ 𝐴 = (𝐹‘𝐵)    ⇒   ⊢ 𝐴 ∈ V
Theorem	fvexd 6771	The value of a class exists (as consequent of anything). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → (𝐹‘𝐴) ∈ V)
Theorem	fvif 6772	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 6773	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 6774*	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 6775	The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
Theorem	fvresd 6776	The value of a restricted function, deduction version of fvres 6775. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
Theorem	funssfv 6777	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	tz6.12-1 6778*	Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦)
Theorem	tz6.12 6779*	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 6780*	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	tz6.12c 6781*	Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
Theorem	tz6.12i 6782	Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.)
⊢ (𝐵 ≠ ∅ → ((𝐹‘𝐴) = 𝐵 → 𝐴𝐹𝐵))
Theorem	fvbr0 6783	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 6784	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	fvssunirn 6785	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.)
⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹
Theorem	ndmfv 6786	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 6787	Reverse closure law for function with the empty set not in its domain. (Contributed by NM, 26-Apr-1996.)
⊢ dom 𝐹 = 𝑆    &   ⊢ ¬ ∅ ∈ 𝑆    ⇒   ⊢ ((𝐹‘𝐴) ∈ 𝑆 → 𝐴 ∈ 𝑆)
Theorem	elfvdm 6788	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 6789	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 6790	If a function value has a member, then its argument is a set. Deduction form of elfvex 6789. (An artifact of our function value definition.) (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (𝐵‘𝐶))    ⇒   ⊢ (𝜑 → 𝐶 ∈ V)
Theorem	eliman0 6791	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 6792	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 6793	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 6794	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 6795	Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
⊢ (∅‘𝐴) = ∅
Theorem	fv2prc 6796	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 6797	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 6798	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 6799	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 6800*	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))