P. 53 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 5201-5300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	forn 5201	The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
Theorem	dffo2 5202	Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵))
Theorem	foima 5203	The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)
Theorem	dffn4 5204	A function maps onto its range. (Contributed by NM, 10-May-1998.)
⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹)
Theorem	funforn 5205	A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴)
Theorem	fodmrnu 5206	An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	fores 5207	Restriction of a function. (Contributed by NM, 4-Mar-1997.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴))
Theorem	foco 5208	Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶)
Theorem	f1oeq1 5209	Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵))
Theorem	f1oeq2 5210	Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶))
Theorem	f1oeq3 5211	Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:𝐶–1-1-onto→𝐵))
Theorem	f1oeq23 5212	Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐷))
Theorem	f1eq123d 5213	Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷))
Theorem	foeq123d 5214	Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐹:𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐷))
Theorem	f1oeq123d 5215	Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐷))
Theorem	nff1o 5216	Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐹:𝐴–1-1-onto→𝐵
Theorem	f1of1 5217	A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵)
Theorem	f1of 5218	A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
Theorem	f1ofn 5219	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 5220	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 5221	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 5222	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 5223	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 5224	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 5225	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 5226	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 5227	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 5228	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 5229	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	f1oabexg 5230*	The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)
⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)}    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V)
Theorem	f1ocnv 5231	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 5232	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 5233	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 5234	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 5235	Preimage of an image. (Contributed by NM, 30-Sep-2004.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝐶)) = 𝐶)
Theorem	foimacnv 5236	A reverse version of f1imacnv 5235. (Contributed by Jeff Hankins, 16-Jul-2009.)
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝐶)) = 𝐶)
Theorem	foun 5237	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 5238	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	fun11iun 5239*	The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.) (Revised by Mario Carneiro, 24-Jun-2015.)
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (∀𝑥 ∈ 𝐴 (𝐵:𝐷–1-1→𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → ∪ 𝑥 ∈ 𝐴 𝐵:∪ 𝑥 ∈ 𝐴 𝐷–1-1→𝑆)
Theorem	resdif 5240	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	f1oco 5241	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 5242	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 5243	Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
Theorem	fococnv2 5244	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 5245	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 5246	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 5247	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 5248	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 5249	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	ffoss 5250*	Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)
⊢ 𝐹 ∈ V    ⇒   ⊢ (𝐹:𝐴⟶𝐵 ↔ ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
Theorem	f11o 5251*	Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)
⊢ 𝐹 ∈ V    ⇒   ⊢ (𝐹:𝐴–1-1→𝐵 ↔ ∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵))
Theorem	f10 5252	The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
⊢ ∅:∅–1-1→𝐴
Theorem	f1o00 5253	One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Theorem	fo00 5254	Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
⊢ (𝐹:∅–onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅))
Theorem	f1o0 5255	One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
⊢ ∅:∅–1-1-onto→∅
Theorem	f1oi 5256	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 5257	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 5258	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 5259	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	f1oprg 5260	An unordered pair of ordered pairs with different elements is a one-to-one onto function. (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 5261*	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 5262*	The value of a function at a unique point. (Contributed by Scott Fenton, 6-Oct-2017.)
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥})
Theorem	brprcneu 5263*	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 5264	A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)
⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
Theorem	fv2 5265*	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	dffv3g 5266*	A definition of function value in terms of iota. (Contributed by Jim Kingdon, 29-Dec-2018.)
⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
Theorem	dffv4g 5267*	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 4770), this definition apparently does not appear in the literature. (Contributed by NM, 1-Aug-1994.)
⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}})
Theorem	elfv 5268*	Membership in a function value. (Contributed by NM, 30-Apr-2004.)
⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥)))
Theorem	fveq1 5269	Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
⊢ (𝐹 = 𝐺 → (𝐹‘𝐴) = (𝐺‘𝐴))
Theorem	fveq2 5270	Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
⊢ (𝐴 = 𝐵 → (𝐹‘𝐴) = (𝐹‘𝐵))
Theorem	fveq1i 5271	Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
⊢ 𝐹 = 𝐺    ⇒   ⊢ (𝐹‘𝐴) = (𝐺‘𝐴)
Theorem	fveq1d 5272	Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
⊢ (𝜑 → 𝐹 = 𝐺)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))
Theorem	fveq2i 5273	Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐹‘𝐴) = (𝐹‘𝐵)
Theorem	fveq2d 5274	Equality deduction for function value. (Contributed by NM, 29-May-1999.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵))
Theorem	2fveq3 5275	Equality theorem for nested function values. (Contributed by AV, 14-Aug-2022.)
⊢ (𝐴 = 𝐵 → (𝐹‘(𝐺‘𝐴)) = (𝐹‘(𝐺‘𝐵)))
Theorem	fveq12i 5276	Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
⊢ 𝐹 = 𝐺    &   ⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐹‘𝐴) = (𝐺‘𝐵)
Theorem	fveq12d 5277	Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
⊢ (𝜑 → 𝐹 = 𝐺)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐵))
Theorem	fveqeq2d 5278	Equality deduction for function value. (Contributed by BJ, 30-Aug-2022.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ((𝐹‘𝐴) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶))
Theorem	fveqeq2 5279	Equality deduction for function value. (Contributed by BJ, 31-Aug-2022.)
⊢ (𝐴 = 𝐵 → ((𝐹‘𝐴) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶))
Theorem	nffv 5280	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 5281*	Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐶)
Theorem	nffvd 5282	Deduction version of bound-variable hypothesis builder nffv 5280. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐹)    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥(𝐹‘𝐴))
Theorem	funfveu 5283*	A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦)
Theorem	fvss 5284*	The value of a function is a subset of 𝐵 if every element that could be a candidate for the value is a subset of 𝐵. (Contributed by Mario Carneiro, 24-May-2019.)
⊢ (∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ 𝐵) → (𝐹‘𝐴) ⊆ 𝐵)
Theorem	fvssunirng 5285	The result of a function value is always a subset of the union of the range, if the input is a set. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
⊢ (𝐴 ∈ V → (𝐹‘𝐴) ⊆ ∪ ran 𝐹)
Theorem	relfvssunirn 5286	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, 24-May-2019.)
⊢ (Rel 𝐹 → (𝐹‘𝐴) ⊆ ∪ ran 𝐹)
Theorem	funfvex 5287	The value of a function exists. A special case of Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by Jim Kingdon, 29-Dec-2018.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ V)
Theorem	relrnfvex 5288	If a function has a set range, then the function value exists unconditional on the domain. (Contributed by Mario Carneiro, 24-May-2019.)
⊢ ((Rel 𝐹 ∧ ran 𝐹 ∈ V) → (𝐹‘𝐴) ∈ V)
Theorem	fvexg 5289	Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐹‘𝐴) ∈ V)
Theorem	fvex 5290	Evaluating a set function at a set exists. (Contributed by Mario Carneiro and Jim Kingdon, 28-May-2019.)
⊢ 𝐹 ∈ 𝑉    &   ⊢ 𝐴 ∈ 𝑊    ⇒   ⊢ (𝐹‘𝐴) ∈ V
Theorem	sefvex 5291	If a function is set-like, then the function value exists if the input does. (Contributed by Mario Carneiro, 24-May-2019.)
⊢ ((◡𝐹 Se V ∧ 𝐴 ∈ V) → (𝐹‘𝐴) ∈ V)
Theorem	fvifdc 5292	Move a conditional outside of a function. (Contributed by Jim Kingdon, 1-Jan-2022.)
⊢ (DECID 𝜑 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵)))
Theorem	fv3 5293*	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 5294	The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
Theorem	funssfv 5295	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 5296*	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 5297*	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 5298*	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 5299*	Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
Theorem	ndmfvg 5300	The value of a class outside its domain is the empty set. (Contributed by Jim Kingdon, 15-Jan-2019.)
⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∅)