P. 71 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 7001-7100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fnsnfv 7001	Singleton of function value. (Contributed by NM, 22-May-1998.) (Proof shortened by Scott Fenton, 8-Aug-2024.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵}))
Theorem	opabiotafun 7002*	Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 19-May-2015.)
⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}    ⇒   ⊢ Fun 𝐹
Theorem	opabiotadm 7003*	Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 16-Nov-2013.)
⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}    ⇒   ⊢ dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
Theorem	opabiota 7004*	Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 16-Nov-2013.)
⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}    &   ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = (℩𝑦𝜓))
Theorem	fnimapr 7005	The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹‘𝐵), (𝐹‘𝐶)})
Theorem	fnimatpd 7006	The image of an unordered triple under a function. (Contributed by Thierry Arnoux, 19-Sep-2023.)
⊢ (𝜑 → 𝐹 Fn 𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹‘𝐴), (𝐹‘𝐵), (𝐹‘𝐶)})
Theorem	ssimaex 7007*	The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))
Theorem	ssimaexg 7008*	The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
⊢ ((𝐴 ∈ 𝐶 ∧ Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥)))
Theorem	funfv 7009	A simplified expression for the value of a function when we know it is a function. (Contributed by NM, 22-May-1998.)
⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))
Theorem	funfv2 7010*	The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by NM, 22-May-1998.)
⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})
Theorem	funfv2f 7011	The value of a function. Version of funfv2 7010 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)
⊢ Ⅎ𝑦𝐴    &   ⊢ Ⅎ𝑦𝐹    ⇒   ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})
Theorem	fvun 7012	Value of the union of two functions when the domains are separate. (Contributed by FL, 7-Nov-2011.)
⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺)‘𝐴) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))
Theorem	fvun1 7013	The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋))
Theorem	fvun2 7014	The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ((𝐴 ∩ 𝐵) = ∅ ∧ 𝑋 ∈ 𝐵)) → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))
Theorem	fvun1d 7015	The value of a union when the argument is in the first domain, a deduction version. (Contributed by metakunt, 28-May-2024.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐹‘𝑋))
Theorem	fvun2d 7016	The value of a union when the argument is in the second domain, a deduction version. (Contributed by metakunt, 28-May-2024.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐵)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐹 ∪ 𝐺)‘𝑋) = (𝐺‘𝑋))
Theorem	dffv2 7017	Alternate definition of function value df-fv 6581 that doesn't require dummy variables. (Contributed by NM, 4-Aug-2010.)
⊢ (𝐹‘𝐴) = ∪ ((𝐹 “ {𝐴}) ∖ ∪ ∪ (((𝐹 ↾ {𝐴}) ∘ ◡(𝐹 ↾ {𝐴})) ∖ I ))
Theorem	dmfco 7018	Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝐴) ∈ dom 𝐹))
Theorem	fvco2 7019	Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (Contributed by NM, 9-Oct-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear, 16-Oct-2014.)
⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
Theorem	fvco 7020	Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario Carneiro, 26-Dec-2014.)
⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐹 ∘ 𝐺)‘𝐴) = (𝐹‘(𝐺‘𝐴)))
Theorem	fvco3 7021	Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)
⊢ ((𝐺:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
Theorem	fvco3d 7022	Value of a function composition. Deduction form of fvco3 7021. (Contributed by Stanislas Polu, 9-Mar-2020.)
⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘𝐶) = (𝐹‘(𝐺‘𝐶)))
Theorem	fvco4i 7023	Conditions for a composition to be expandable without conditions on the argument. (Contributed by Stefan O'Rear, 31-Mar-2015.)
⊢ ∅ = (𝐹‘∅)    &   ⊢ Fun 𝐺    ⇒   ⊢ ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋))
Theorem	fvopab3g 7024*	Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 ∈ 𝐶 → ∃!𝑦𝜑)    &   ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ((𝐹‘𝐴) = 𝐵 ↔ 𝜒))
Theorem	fvopab3ig 7025*	Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 ∈ 𝐶 → ∃*𝑦𝜑)    &   ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝜑)}    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝜒 → (𝐹‘𝐴) = 𝐵))
Theorem	brfvopabrbr 7026*	The binary relation of a function value which is an ordered-pair class abstraction of a restricted binary relation is the restricted binary relation. The first hypothesis can often be obtained by using fvmptopab 7504. (Contributed by AV, 29-Oct-2021.)
⊢ (𝐴‘𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐵‘𝑍)𝑦 ∧ 𝜑)}    &   ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝜑 ↔ 𝜓))    &   ⊢ Rel (𝐵‘𝑍)    ⇒   ⊢ (𝑋(𝐴‘𝑍)𝑌 ↔ (𝑋(𝐵‘𝑍)𝑌 ∧ 𝜓))
Theorem	fvmptg 7027*	Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    ⇒   ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅) → (𝐹‘𝐴) = 𝐶)
Theorem	fvmpti 7028*	Value of a function given in maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    ⇒   ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = ( I ‘𝐶))
Theorem	fvmpt 7029*	Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶)
Theorem	fvmpt2f 7030	Value of a function given by the maps-to notation. (Contributed by Thierry Arnoux, 9-Mar-2017.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
Theorem	fvtresfn 7031*	Functionality of a tuple-restriction function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉))    ⇒   ⊢ (𝑋 ∈ 𝐵 → (𝐹‘𝑋) = (𝑋 ↾ 𝑉))
Theorem	fvmpts 7032*	Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ 𝐵)    ⇒   ⊢ ((𝐴 ∈ 𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵)
Theorem	fvmpt3 7033*	Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    &   ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶)
Theorem	fvmpt3i 7034*	Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶)
Theorem	fvmptdf 7035*	Deduction version of fvmptd 7036 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by AV, 29-Mar-2024.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐶    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)
Theorem	fvmptd 7036*	Deduction version of fvmpt 7029. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) (Proof shortened by AV, 29-Mar-2024.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)
Theorem	fvmptd2 7037*	Deduction version of fvmpt 7029 (where the definition of the mapping does not depend on the common antecedent 𝜑). (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)
Theorem	mptrcl 7038*	Reverse closure for a mapping: If the function value of a mapping has a member, the argument belongs to the base class of the mapping. (Contributed by AV, 4-Apr-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝐼 ∈ (𝐹‘𝑋) → 𝑋 ∈ 𝐴)
Theorem	fvmpt2i 7039*	Value of a function given by the maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = ( I ‘𝐵))
Theorem	fvmpt2 7040*	Value of a function given by the maps-to notation. (Contributed by FL, 21-Jun-2010.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐹‘𝑥) = 𝐵)
Theorem	fvmptss 7041*	If all the values of the mapping are subsets of a class 𝐶, then so is any evaluation of the mapping, even if 𝐷 is not in the base set 𝐴. (Contributed by Mario Carneiro, 13-Feb-2015.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶)
Theorem	fvmpt2d 7042*	Deduction version of fvmpt2 7040. (Contributed by Thierry Arnoux, 8-Dec-2016.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
Theorem	fvmptex 7043*	Express a function 𝐹 whose value 𝐵 may not always be a set in terms of another function 𝐺 for which sethood is guaranteed. (Note that ( I ‘𝐵) is just shorthand for if(𝐵 ∈ V, 𝐵, ∅), and it is always a set by fvex 6933.) Note also that these functions are not the same; wherever 𝐵(𝐶) is not a set, 𝐶 is not in the domain of 𝐹 (so it evaluates to the empty set), but 𝐶 is in the domain of 𝐺, and 𝐺(𝐶) is defined to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵))    ⇒   ⊢ (𝐹‘𝐶) = (𝐺‘𝐶)
Theorem	fvmptd3f 7044*	Alternate deduction version of fvmpt 7029 with three nonfreeness hypotheses instead of distinct variable conditions. (Contributed by AV, 19-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓))    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓))
Theorem	fvmptd2f 7045*	Alternate deduction version of fvmpt 7029, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) (Proof shortened by AV, 19-Jan-2022.)
⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓))    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓))
Theorem	fvmptdv 7046*	Alternate deduction version of fvmpt 7029, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓))    ⇒   ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓))
Theorem	fvmptdv2 7047*	Alternate deduction version of fvmpt 7029, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = 𝐶))
Theorem	mpteqb 7048*	Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 7064. (Contributed by Mario Carneiro, 14-Nov-2014.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶))
Theorem	fvmptt 7049*	Closed theorem form of fvmpt 7029. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
⊢ ((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → (𝐹‘𝐴) = 𝐶)
Theorem	fvmptf 7050*	Value of a function given by an ordered-pair class abstraction. This version of fvmptg 7027 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    ⇒   ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶)
Theorem	fvmptnf 7051*	The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvmptn 7054 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    ⇒   ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅)
Theorem	fvmptd3 7052*	Deduction version of fvmpt 7029. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)
Theorem	fvmptd4 7053*	Deduction version of fvmpt 7029 (where the substitution hypothesis does not have the antecedent 𝜑). (Contributed by SN, 26-Jul-2024.)
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵))    &   ⊢ (𝜑 → 𝐴 ∈ 𝐷)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶)
Theorem	fvmptn 7054*	This somewhat non-intuitive theorem tells us the value of its function is the empty set when the class 𝐶 it would otherwise map to is a proper class. This is a technical lemma that can help eliminate redundant sethood antecedents otherwise required by fvmptg 7027. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 9-Sep-2013.)
⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐷) = ∅)
Theorem	fvmptss2 7055*	A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.)
⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝐹‘𝐷) ⊆ 𝐶
Theorem	elfvmptrab1w 7056*	Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. Version of elfvmptrab1 7057 with a disjoint variable condition, which does not require ax-13 2380. (Contributed by Alexander van der Vekens, 15-Jul-2018.) Avoid ax-13 2380. (Revised by GG, 26-Jan-2024.)
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})    &   ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)    ⇒   ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))
Theorem	elfvmptrab1 7057*	Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. Usage of this theorem is discouraged because it depends on ax-13 2380. Use the weaker elfvmptrab1w 7056 when possible. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (New usage is discouraged.)
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑})    &   ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V)    ⇒   ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀))
Theorem	elfvmptrab 7058*	Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑})    &   ⊢ (𝑋 ∈ 𝑉 → 𝑀 ∈ V)    ⇒   ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑀))
Theorem	fvopab4ndm 7059*	Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)
⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}    ⇒   ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅)
Theorem	fvmptndm 7060*	Value of a function given by the maps-to notation, outside of its domain. (Contributed by AV, 31-Dec-2020.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (¬ 𝑋 ∈ 𝐴 → (𝐹‘𝑋) = ∅)
Theorem	fvmptrabfv 7061*	Value of a function mapping a set to a class abstraction restricting the value of another function. (Contributed by AV, 18-Feb-2022.)
⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑})    &   ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓}
Theorem	fvopab5 7062*	The value of a function that is expressed as an ordered pair abstraction. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 11-Sep-2015.)
⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑦𝜓))
Theorem	fvopab6 7063*	Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐵)}    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶)    ⇒   ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅 ∧ 𝜓) → (𝐹‘𝐴) = 𝐶)
Theorem	eqfnfv 7064*	Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
Theorem	eqfnfv2 7065*	Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))))
Theorem	eqfnfv3 7066*	Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥)))))
Theorem	eqfnfvd 7067*	Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐺 Fn 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥))    ⇒   ⊢ (𝜑 → 𝐹 = 𝐺)
Theorem	eqfnfv2f 7068*	Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 7064 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐺    ⇒   ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))
Theorem	eqfunfv 7069*	Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)
⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = (𝐺‘𝑥))))
Theorem	eqfnun 7070	Two functions on 𝐴 ∪ 𝐵 are equal if and only if they have equal restrictions to both 𝐴 and 𝐵. (Contributed by Jeff Madsen, 19-Jun-2011.)
⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (𝐹 = 𝐺 ↔ ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵))))
Theorem	fvreseq0 7071*	Equality of restricted functions is determined by their values (for functions with different domains). (Contributed by AV, 6-Jan-2019.)
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐶)) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
Theorem	fvreseq1 7072*	Equality of a function restricted to the domain of another function. (Contributed by AV, 6-Jan-2019.)
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ↾ 𝐵) = 𝐺 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
Theorem	fvreseq 7073*	Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.) (Proof shortened by AV, 4-Mar-2019.)
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
Theorem	fnmptfvd 7074*	A function with a given domain is a mapping defined by its function values. (Contributed by AV, 1-Mar-2019.)
⊢ (𝜑 → 𝑀 Fn 𝐴)    &   ⊢ (𝑖 = 𝑎 → 𝐷 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝐷 ∈ 𝑈)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷))
Theorem	fndmdif 7075*	Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)})
Theorem	fndmdifcom 7076	The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = dom (𝐺 ∖ 𝐹))
Theorem	fndmdifeq0 7077	The difference set of two functions is empty if and only if the functions are equal. (Contributed by Stefan O'Rear, 17-Jan-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom (𝐹 ∖ 𝐺) = ∅ ↔ 𝐹 = 𝐺))
Theorem	fndmin 7078*	Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∩ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)})
Theorem	fneqeql 7079	Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ dom (𝐹 ∩ 𝐺) = 𝐴))
Theorem	fneqeql2 7080	Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ 𝐴 ⊆ dom (𝐹 ∩ 𝐺)))
Theorem	fnreseql 7081	Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝐹 ∩ 𝐺)))
Theorem	chfnrn 7082*	The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by NM, 31-Aug-1999.)
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥) → ran 𝐹 ⊆ ∪ 𝐴)
Theorem	funfvop 7083	Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹)
Theorem	funfvbrb 7084	Two ways to say that 𝐴 is in the domain of 𝐹. (Contributed by Mario Carneiro, 1-May-2014.)
⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 ↔ 𝐴𝐹(𝐹‘𝐴)))
Theorem	fvimacnvi 7085	A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → (𝐹‘𝐴) ∈ 𝐵)
Theorem	fvimacnv 7086	The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 6661 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))
Theorem	funimass3 7087	A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 7086 would be the special case of 𝐴 being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵)))
Theorem	funimass5 7088*	A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐴 ⊆ (◡𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
Theorem	funconstss 7089*	Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ {𝐵})))
Theorem	fvimacnvALT 7090	Alternate proof of fvimacnv 7086, based on funimass3 7087. If funimass3 7087 is ever proved directly, as opposed to using funimacnv 6659 pointwise, then the proof of funimacnv 6659 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵)))
Theorem	elpreima 7091	Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ (◡𝐹 “ 𝐶) ↔ (𝐵 ∈ 𝐴 ∧ (𝐹‘𝐵) ∈ 𝐶)))
Theorem	elpreimad 7092	Membership in the preimage of a set under a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐹‘𝐵) ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 ∈ (◡𝐹 “ 𝐶))
Theorem	fniniseg 7093	Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.) (Proof shortened by Mario Carneiro , 28-Apr-2015.)
⊢ (𝐹 Fn 𝐴 → (𝐶 ∈ (◡𝐹 “ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) = 𝐵)))
Theorem	fncnvima2 7094*	Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ 𝐵) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝐵})
Theorem	fniniseg2 7095*	Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝐵})
Theorem	unpreima 7096	Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵)))
Theorem	inpreima 7097	Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)
⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵)))
Theorem	difpreima 7098	Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∖ 𝐵)) = ((◡𝐹 “ 𝐴) ∖ (◡𝐹 “ 𝐵)))
Theorem	respreima 7099	The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐵) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ 𝐵))
Theorem	cnvimainrn 7100	The preimage of the intersection of the range of a class and a class 𝐴 is the preimage of the class 𝐴. (Contributed by AV, 17-Sep-2024.)
⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = (◡𝐹 “ 𝐴))