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

Theoremf1oeq3d 6601 Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹:𝐶1-1-onto𝐴𝐹:𝐶1-1-onto𝐵))

Theoremnff1o 6602 Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)
𝑥𝐹    &   𝑥𝐴    &   𝑥𝐵       𝑥 𝐹:𝐴1-1-onto𝐵

Theoremf1of1 6603 A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵𝐹:𝐴1-1𝐵)

Theoremf1of 6604 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)

Theoremf1ofn 6605 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵𝐹 Fn 𝐴)

Theoremf1ofun 6606 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)

Theoremf1orel 6607 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)
(𝐹:𝐴1-1-onto𝐵 → Rel 𝐹)

Theoremf1odm 6608 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)
(𝐹:𝐴1-1-onto𝐵 → dom 𝐹 = 𝐴)

Theoremdff1o2 6609 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹 ∧ ran 𝐹 = 𝐵))

Theoremdff1o3 6610 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴onto𝐵 ∧ Fun 𝐹))

Theoremf1ofo 6611 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)
(𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)

Theoremdff1o4 6612 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))

Theoremdff1o5 6613 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
(𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴1-1𝐵 ∧ ran 𝐹 = 𝐵))

Theoremf1orn 6614 A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)
(𝐹:𝐴1-1-onto→ran 𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun 𝐹))

Theoremf1f1orn 6615 A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)
(𝐹:𝐴1-1𝐵𝐹:𝐴1-1-onto→ran 𝐹)

Theoremf1ocnv 6616 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:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)

Theoremf1ocnvb 6617 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:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴))

Theoremf1ores 6618 The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)
((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹𝐶):𝐶1-1-onto→(𝐹𝐶))

Theoremf1orescnv 6619 The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)
((Fun 𝐹 ∧ (𝐹𝑅):𝑅1-1-onto𝑃) → (𝐹𝑃):𝑃1-1-onto𝑅)

Theoremf1imacnv 6620 Preimage of an image. (Contributed by NM, 30-Sep-2004.)
((𝐹:𝐴1-1𝐵𝐶𝐴) → (𝐹 “ (𝐹𝐶)) = 𝐶)

Theoremfoimacnv 6621 A reverse version of f1imacnv 6620. (Contributed by Jeff Hankins, 16-Jul-2009.)
((𝐹:𝐴onto𝐵𝐶𝐵) → (𝐹 “ (𝐹𝐶)) = 𝐶)

Theoremfoun 6622 The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)
(((𝐹:𝐴onto𝐵𝐺:𝐶onto𝐷) ∧ (𝐴𝐶) = ∅) → (𝐹𝐺):(𝐴𝐶)–onto→(𝐵𝐷))

Theoremf1oun 6623 The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)
(((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷) ∧ ((𝐴𝐶) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝐹𝐺):(𝐴𝐶)–1-1-onto→(𝐵𝐷))

Theoremresdif 6624 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 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))

Theoremresin 6625 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 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))

Theoremf1oco 6626 Composition of one-to-one onto functions. (Contributed by NM, 19-Mar-1998.)
((𝐹:𝐵1-1-onto𝐶𝐺:𝐴1-1-onto𝐵) → (𝐹𝐺):𝐴1-1-onto𝐶)

Theoremf1cnv 6627 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)
(𝐹:𝐴1-1𝐵𝐹:ran 𝐹1-1-onto𝐴)

Theoremfuncocnv2 6628 Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → (𝐹𝐹) = ( I ↾ ran 𝐹))

Theoremfococnv2 6629 The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
(𝐹:𝐴onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))

Theoremf1ococnv2 6630 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:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐵))

Theoremf1cocnv2 6631 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
(𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ ran 𝐹))

Theoremf1ococnv1 6632 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:𝐴1-1-onto𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))

Theoremf1cocnv1 6633 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
(𝐹:𝐴1-1𝐵 → (𝐹𝐹) = ( I ↾ 𝐴))

Theoremfuncoeqres 6634 Express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)
((Fun 𝐺 ∧ (𝐹𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻𝐺))

Theoremf1ssf1 6635 A subset of an injective function is injective. (Contributed by AV, 20-Nov-2020.)
((Fun 𝐹 ∧ Fun 𝐹𝐺𝐹) → Fun 𝐺)

Theoremf10 6636 The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
∅:∅–1-1𝐴

Theoremf10d 6637 The empty set maps one-to-one into any class, deduction version. (Contributed by AV, 25-Nov-2020.)
(𝜑𝐹 = ∅)       (𝜑𝐹:dom 𝐹1-1𝐴)

Theoremf1o00 6638 One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
(𝐹:∅–1-1-onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))

Theoremfo00 6639 Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
(𝐹:∅–onto𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))

Theoremf1o0 6640 One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
∅:∅–1-1-onto→∅

Theoremf1oi 6641 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):𝐴1-1-onto𝐴

Theoremf1ovi 6642 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

Theoremf1osn 6643 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}

Theoremf1osng 6644 A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1-onto→{𝐵})

Theoremf1sng 6645 A singleton of an ordered pair is a one-to-one function. (Contributed by AV, 17-Apr-2021.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩}:{𝐴}–1-1𝑊)

Theoremfsnd 6646 A singleton of an ordered pair is a function. (Contributed by AV, 17-Apr-2021.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)       (𝜑 → {⟨𝐴, 𝐵⟩}:{𝐴}⟶𝑊)

Theoremf1oprswap 6647 A two-element swap is a bijection on a pair. (Contributed by Mario Carneiro, 23-Jan-2015.)
((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩, ⟨𝐵, 𝐴⟩}:{𝐴, 𝐵}–1-1-onto→{𝐴, 𝐵})

Theoremf1oprg 6648 An unordered pair of ordered pairs with different elements is a one-to-one onto function, analogous to f1oprswap 6647. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑋𝐷𝑌)) → ((𝐴𝐶𝐵𝐷) → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}:{𝐴, 𝐶}–1-1-onto→{𝐵, 𝐷}))

Theoremtz6.12-2 6649* 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 → (𝐹𝐴) = ∅)

Theoremfveu 6650* The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
(∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = {𝑥𝐴𝐹𝑥})

Theorembrprcneu 6651* 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)

Theoremfvprc 6652 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)
𝐴 ∈ V → (𝐹𝐴) = ∅)

Theoremrnfvprc 6653 The range of a function value at a proper class is empty. (Contributed by AV, 20-Aug-2022.)
𝑌 = (𝐹𝑋)       𝑋 ∈ V → ran 𝑌 = ∅)

Theoremfv2 6654* 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.)
(𝐹𝐴) = {𝑥 ∣ ∀𝑦(𝐴𝐹𝑦𝑦 = 𝑥)}

Theoremdffv3 6655* A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.)
(𝐹𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))

Theoremdffv4 6656* 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 5945), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
(𝐹𝐴) = {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}

Theoremelfv 6657* Membership in a function value. (Contributed by NM, 30-Apr-2004.)
(𝐴 ∈ (𝐹𝐵) ↔ ∃𝑥(𝐴𝑥 ∧ ∀𝑦(𝐵𝐹𝑦𝑦 = 𝑥)))

Theoremfveq1 6658 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
(𝐹 = 𝐺 → (𝐹𝐴) = (𝐺𝐴))

Theoremfveq2 6659 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
(𝐴 = 𝐵 → (𝐹𝐴) = (𝐹𝐵))

Theoremfveq1i 6660 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
𝐹 = 𝐺       (𝐹𝐴) = (𝐺𝐴)

Theoremfveq1d 6661 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
(𝜑𝐹 = 𝐺)       (𝜑 → (𝐹𝐴) = (𝐺𝐴))

Theoremfveq2i 6662 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
𝐴 = 𝐵       (𝐹𝐴) = (𝐹𝐵)

Theoremfveq2d 6663 Equality deduction for function value. (Contributed by NM, 29-May-1999.)
(𝜑𝐴 = 𝐵)       (𝜑 → (𝐹𝐴) = (𝐹𝐵))

Theorem2fveq3 6664 Equality theorem for nested function values. (Contributed by AV, 14-Aug-2022.)
(𝐴 = 𝐵 → (𝐹‘(𝐺𝐴)) = (𝐹‘(𝐺𝐵)))

Theoremfveq12i 6665 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
𝐹 = 𝐺    &   𝐴 = 𝐵       (𝐹𝐴) = (𝐺𝐵)

Theoremfveq12d 6666 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
(𝜑𝐹 = 𝐺)    &   (𝜑𝐴 = 𝐵)       (𝜑 → (𝐹𝐴) = (𝐺𝐵))

Theoremfveqeq2d 6667 Equality deduction for function value. (Contributed by BJ, 30-Aug-2022.)
(𝜑𝐴 = 𝐵)       (𝜑 → ((𝐹𝐴) = 𝐶 ↔ (𝐹𝐵) = 𝐶))

Theoremfveqeq2 6668 Equality deduction for function value. (Contributed by BJ, 31-Aug-2022.)
(𝐴 = 𝐵 → ((𝐹𝐴) = 𝐶 ↔ (𝐹𝐵) = 𝐶))

Theoremnffv 6669 Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐹    &   𝑥𝐴       𝑥(𝐹𝐴)

Theoremnffvmpt1 6670* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
𝑥((𝑥𝐴𝐵)‘𝐶)

Theoremnffvd 6671 Deduction version of bound-variable hypothesis builder nffv 6669. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
(𝜑𝑥𝐹)    &   (𝜑𝑥𝐴)       (𝜑𝑥(𝐹𝐴))

Theoremfvex 6672 The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by NM, 30-Dec-1996.)
(𝐹𝐴) ∈ V

Theoremfvexi 6673 The value of a class exists. Inference form of fvex 6672. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
𝐴 = (𝐹𝐵)       𝐴 ∈ V

Theoremfvexd 6674 The value of a class exists (as consequent of anything). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
(𝜑 → (𝐹𝐴) ∈ V)

Theoremfvif 6675 Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹𝐴), (𝐹𝐵))

Theoremiffv 6676 Move a conditional outside of a function. (Contributed by Thierry Arnoux, 28-Sep-2018.)
(if(𝜑, 𝐹, 𝐺)‘𝐴) = if(𝜑, (𝐹𝐴), (𝐺𝐴))

Theoremfv3 6677* 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.)
(𝐹𝐴) = {𝑥 ∣ (∃𝑦(𝑥𝑦𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)}

Theoremfvres 6678 The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
(𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))

Theoremfvresd 6679 The value of a restricted function, deduction version of fvres 6678. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
(𝜑𝐴𝐵)       (𝜑 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))

Theoremfunssfv 6680 The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.)
((Fun 𝐹𝐺𝐹𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))

Theoremtz6.12-1 6681* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)

Theoremtz6.12 6682* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)
((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)

Theoremtz6.12f 6683* Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)
𝑦𝐹       ((⟨𝐴, 𝑦⟩ ∈ 𝐹 ∧ ∃!𝑦𝐴, 𝑦⟩ ∈ 𝐹) → (𝐹𝐴) = 𝑦)

Theoremtz6.12c 6684* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
(∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))

Theoremtz6.12i 6685 Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.)
(𝐵 ≠ ∅ → ((𝐹𝐴) = 𝐵𝐴𝐹𝐵))

Theoremfvbr0 6686 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(𝐹𝑋) ∨ (𝐹𝑋) = ∅)

Theoremfvrn0 6687 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.)
(𝐹𝑋) ∈ (ran 𝐹 ∪ {∅})

Theoremfvssunirn 6688 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.)
(𝐹𝑋) ⊆ ran 𝐹

Theoremndmfv 6689 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 → (𝐹𝐴) = ∅)

Theoremndmfvrcl 6690 Reverse closure law for function with the empty set not in its domain. (Contributed by NM, 26-Apr-1996.)
dom 𝐹 = 𝑆    &    ¬ ∅ ∈ 𝑆       ((𝐹𝐴) ∈ 𝑆𝐴𝑆)

Theoremelfvdm 6691 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 ∈ (𝐹𝐵) → 𝐵 ∈ dom 𝐹)

Theoremelfvex 6692 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 ∈ (𝐹𝐵) → 𝐵 ∈ V)

Theoremelfvexd 6693 If a function value has a member, then its argument is a set. Deduction form of elfvex 6692. (An artifact of our function value definition.) (Contributed by David Moews, 1-May-2017.)
(𝜑𝐴 ∈ (𝐵𝐶))       (𝜑𝐶 ∈ V)

Theoremeliman0 6694 A nonempty function value is an element of the image of the function. (Contributed by Thierry Arnoux, 25-Jun-2019.)
((𝐴𝐵 ∧ ¬ (𝐹𝐴) = ∅) → (𝐹𝐴) ∈ (𝐹𝐵))

Theoremnfvres 6695 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.)
𝐴𝐵 → ((𝐹𝐵)‘𝐴) = ∅)

Theoremnfunsn 6696 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}) → (𝐹𝐴) = ∅)

Theoremfvfundmfvn0 6697 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.)
((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))

Theorem0fv 6698 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
(∅‘𝐴) = ∅

Theoremfv2prc 6699 A function value of a function value at a proper class is the empty set. (Contributed by AV, 8-Apr-2021.)
𝐴 ∈ V → ((𝐹𝐴)‘𝐵) = ∅)

Theoremelfv2ex 6700 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 ∈ ((𝐹𝐵)‘𝐶) → 𝐵 ∈ V)

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