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Theorem List for Metamath Proof Explorer - 6301-6400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfnimapr 6301 The image of a pair under a function. (Contributed by Jeff Madsen, 6-Jan-2011.)
((𝐹 Fn 𝐴𝐵𝐴𝐶𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹𝐵), (𝐹𝐶)})
 
Theoremssimaex 6302* The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
𝐴 ∈ V       ((Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
 
Theoremssimaexg 6303* The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
((𝐴𝐶 ∧ Fun 𝐹𝐵 ⊆ (𝐹𝐴)) → ∃𝑥(𝑥𝐴𝐵 = (𝐹𝑥)))
 
Theoremfunfv 6304 A simplified expression for the value of a function when we know it's a function. (Contributed by NM, 22-May-1998.)
(Fun 𝐹 → (𝐹𝐴) = (𝐹 “ {𝐴}))
 
Theoremfunfv2 6305* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by NM, 22-May-1998.)
(Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
 
Theoremfunfv2f 6306 The value of a function. Version of funfv2 6305 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)
𝑦𝐴    &   𝑦𝐹       (Fun 𝐹 → (𝐹𝐴) = {𝑦𝐴𝐹𝑦})
 
Theoremfvun 6307 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) = ∅) → ((𝐹𝐺)‘𝐴) = ((𝐹𝐴) ∪ (𝐺𝐴)))
 
Theoremfvun1 6308 The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹𝑋))
 
Theoremfvun2 6309 The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ ((𝐴𝐵) = ∅ ∧ 𝑋𝐵)) → ((𝐹𝐺)‘𝑋) = (𝐺𝑋))
 
Theoremdffv2 6310 Alternate definition of function value df-fv 5934 that doesn't require dummy variables. (Contributed by NM, 4-Aug-2010.)
(𝐹𝐴) = ((𝐹 “ {𝐴}) ∖ (((𝐹 ↾ {𝐴}) ∘ (𝐹 ↾ {𝐴})) ∖ I ))
 
Theoremdmfco 6311 Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
((Fun 𝐺𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹𝐺) ↔ (𝐺𝐴) ∈ dom 𝐹))
 
Theoremfvco2 6312 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 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
 
Theoremfvco 6313 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 𝐺𝐴 ∈ dom 𝐺) → ((𝐹𝐺)‘𝐴) = (𝐹‘(𝐺𝐴)))
 
Theoremfvco3 6314 Value of a function composition. (Contributed by NM, 3-Jan-2004.) (Revised by Mario Carneiro, 26-Dec-2014.)
((𝐺:𝐴𝐵𝐶𝐴) → ((𝐹𝐺)‘𝐶) = (𝐹‘(𝐺𝐶)))
 
Theoremfvco4i 6315 Conditions for a composition to be expandable without conditions on the argument. (Contributed by Stefan O'Rear, 31-Mar-2015.)
∅ = (𝐹‘∅)    &   Fun 𝐺       ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋))
 
Theoremfvopab3g 6316* 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 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑥𝐶 → ∃!𝑦𝜑)    &   𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}       ((𝐴𝐶𝐵𝐷) → ((𝐹𝐴) = 𝐵𝜒))
 
Theoremfvopab3ig 6317* Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)
(𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵 → (𝜓𝜒))    &   (𝑥𝐶 → ∃*𝑦𝜑)    &   𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐶𝜑)}       ((𝐴𝐶𝐵𝐷) → (𝜒 → (𝐹𝐴) = 𝐵))
 
Theorembrfvopabrbr 6318* 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 6739. (Contributed by AV, 29-Oct-2021.)
(𝐴𝑍) = {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝐵𝑍)𝑦𝜑)}    &   ((𝑥 = 𝑋𝑦 = 𝑌) → (𝜑𝜓))    &   Rel (𝐵𝑍)       (𝑋(𝐴𝑍)𝑌 ↔ (𝑋(𝐵𝑍)𝑌𝜓))
 
Theoremfvmptg 6319* Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)       ((𝐴𝐷𝐶𝑅) → (𝐹𝐴) = 𝐶)
 
Theoremfvmpti 6320* Value of a function given in maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
(𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)       (𝐴𝐷 → (𝐹𝐴) = ( I ‘𝐶))
 
Theoremfvmpt 6321* Value of a function given in maps-to notation. (Contributed by NM, 17-Aug-2011.)
(𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)    &   𝐶 ∈ V       (𝐴𝐷 → (𝐹𝐴) = 𝐶)
 
Theoremfvmpt2f 6322 Value of a function given by the "maps to" notation. (Contributed by Thierry Arnoux, 9-Mar-2017.)
𝑥𝐴       ((𝑥𝐴𝐵𝐶) → ((𝑥𝐴𝐵)‘𝑥) = 𝐵)
 
Theoremfvtresfn 6323* Functionality of a tuple-restriction function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))       (𝑋𝐵 → (𝐹𝑋) = (𝑋𝑉))
 
Theoremfvmpts 6324* 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 = (𝑥𝐶𝐵)       ((𝐴𝐶𝐴 / 𝑥𝐵𝑉) → (𝐹𝐴) = 𝐴 / 𝑥𝐵)
 
Theoremfvmpt3 6325* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.)
(𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)    &   (𝑥𝐷𝐵𝑉)       (𝐴𝐷 → (𝐹𝐴) = 𝐶)
 
Theoremfvmpt3i 6326* Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.)
(𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)    &   𝐵 ∈ V       (𝐴𝐷 → (𝐹𝐴) = 𝐶)
 
Theoremfvmptd 6327* Deduction version of fvmpt 6321. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
(𝜑𝐹 = (𝑥𝐷𝐵))    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)    &   (𝜑𝐴𝐷)    &   (𝜑𝐶𝑉)       (𝜑 → (𝐹𝐴) = 𝐶)
 
Theoremmptrcl 6328* 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 = (𝑥𝐴𝐵)       (𝐼 ∈ (𝐹𝑋) → 𝑋𝐴)
 
Theoremfvmpt2i 6329* Value of a function given by the "maps to" notation. (Contributed by Mario Carneiro, 23-Apr-2014.)
𝐹 = (𝑥𝐴𝐵)       (𝑥𝐴 → (𝐹𝑥) = ( I ‘𝐵))
 
Theoremfvmpt2 6330* Value of a function given by the "maps to" notation. (Contributed by FL, 21-Jun-2010.)
𝐹 = (𝑥𝐴𝐵)       ((𝑥𝐴𝐵𝐶) → (𝐹𝑥) = 𝐵)
 
Theoremfvmptss 6331* 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 = (𝑥𝐴𝐵)       (∀𝑥𝐴 𝐵𝐶 → (𝐹𝐷) ⊆ 𝐶)
 
Theoremfvmpt2d 6332* Deduction version of fvmpt2 6330. (Contributed by Thierry Arnoux, 8-Dec-2016.)
(𝜑𝐹 = (𝑥𝐴𝐵))    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐵)
 
Theoremfvmptex 6333* 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 6239.) 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 = (𝑥𝐴𝐵)    &   𝐺 = (𝑥𝐴 ↦ ( I ‘𝐵))       (𝐹𝐶) = (𝐺𝐶)
 
Theoremfvmptd3f 6334* Alternate deduction version of fvmpt 6321 with three non-freeness hypotheses instead of distinct variable conditions. (Contributed by AV, 19-Jan-2022.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))    &   𝑥𝐹    &   𝑥𝜓    &   𝑥𝜑       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
 
Theoremfvmptdf 6335* Alternate deduction version of fvmpt 6321, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) (Proof shortened by AV, 19-Jan-2022.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))    &   𝑥𝐹    &   𝑥𝜓       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
 
Theoremfvmptdv 6336* Alternate deduction version of fvmpt 6321, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
 
Theoremfvmptdv2 6337* Alternate deduction version of fvmpt 6321, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
(𝜑𝐴𝐷)    &   ((𝜑𝑥 = 𝐴) → 𝐵𝑉)    &   ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)       (𝜑 → (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = 𝐶))
 
Theoremmpteqb 6338* Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 6351. (Contributed by Mario Carneiro, 14-Nov-2014.)
(∀𝑥𝐴 𝐵𝑉 → ((𝑥𝐴𝐵) = (𝑥𝐴𝐶) ↔ ∀𝑥𝐴 𝐵 = 𝐶))
 
Theoremfvmptt 6339* Closed theorem form of fvmpt 6321. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.)
((∀𝑥(𝑥 = 𝐴𝐵 = 𝐶) ∧ 𝐹 = (𝑥𝐷𝐵) ∧ (𝐴𝐷𝐶𝑉)) → (𝐹𝐴) = 𝐶)
 
Theoremfvmptf 6340* Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6319 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
𝑥𝐴    &   𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)       ((𝐴𝐷𝐶𝑉) → (𝐹𝐴) = 𝐶)
 
Theoremfvmptnf 6341* 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 6342 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
𝑥𝐴    &   𝑥𝐶    &   (𝑥 = 𝐴𝐵 = 𝐶)    &   𝐹 = (𝑥𝐷𝐵)       𝐶 ∈ V → (𝐹𝐴) = ∅)
 
Theoremfvmptn 6342* 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 6319. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 9-Sep-2013.)
(𝑥 = 𝐷𝐵 = 𝐶)    &   𝐹 = (𝑥𝐴𝐵)       𝐶 ∈ V → (𝐹𝐷) = ∅)
 
Theoremfvmptss2 6343* 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 = 𝐷𝐵 = 𝐶)    &   𝐹 = (𝑥𝐴𝐵)       (𝐹𝐷) ⊆ 𝐶
 
Theoremelfvmptrab1 6344* 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. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝑥 / 𝑚𝑀𝜑})    &   (𝑋𝑉𝑋 / 𝑚𝑀 ∈ V)       (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑋 / 𝑚𝑀))
 
Theoremelfvmptrab 6345* 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 = (𝑥𝑉 ↦ {𝑦𝑀𝜑})    &   (𝑋𝑉𝑀 ∈ V)       (𝑌 ∈ (𝐹𝑋) → (𝑋𝑉𝑌𝑀))
 
Theoremfvopab4ndm 6346* Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.)
𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝜑)}       𝐵𝐴 → (𝐹𝐵) = ∅)
 
Theoremfvmptndm 6347* Value of a function given by the "maps to" notation, outside of its domain. (Contributed by AV, 31-Dec-2020.)
𝐹 = (𝑥𝐴𝐵)       𝑋𝐴 → (𝐹𝑋) = ∅)
 
Theoremfvmptrabfv 6348* 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 ∈ (𝐺𝑥) ∣ 𝜑})    &   (𝑥 = 𝑋 → (𝜑𝜓))       (𝐹𝑋) = {𝑦 ∈ (𝐺𝑋) ∣ 𝜓}
 
Theoremfvopab5 6349* 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 → (𝜑𝜓))       (𝐴𝑉 → (𝐹𝐴) = (℩𝑦𝜓))
 
Theoremfvopab6 6350* 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⟩ ∣ (𝜑𝑦 = 𝐵)}    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑥 = 𝐴𝐵 = 𝐶)       ((𝐴𝐷𝐶𝑅𝜓) → (𝐹𝐴) = 𝐶)
 
Theoremeqfnfv 6351* 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 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
 
Theoremeqfnfv2 6352* 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 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥))))
 
Theoremeqfnfv3 6353* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵𝐴 ∧ ∀𝑥𝐴 (𝑥𝐵 ∧ (𝐹𝑥) = (𝐺𝑥)))))
 
Theoremeqfnfvd 6354* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
(𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)    &   ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐺𝑥))       (𝜑𝐹 = 𝐺)
 
Theoremeqfnfv2f 6355* 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 6351 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)
𝑥𝐹    &   𝑥𝐺       ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥𝐴 (𝐹𝑥) = (𝐺𝑥)))
 
Theoremeqfunfv 6356* 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(𝐹𝑥) = (𝐺𝑥))))
 
Theoremfvreseq0 6357* Equality of restricted functions is determined by their values (for functions with different domains). (Contributed by AV, 6-Jan-2019.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐶) ∧ (𝐵𝐴𝐵𝐶)) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
 
Theoremfvreseq1 6358* Equality of a function restricted to the domain of another function. (Contributed by AV, 6-Jan-2019.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐵) ∧ 𝐵𝐴) → ((𝐹𝐵) = 𝐺 ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
 
Theoremfvreseq 6359* Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.) (Proof shortened by AV, 4-Mar-2019.)
(((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ 𝐵𝐴) → ((𝐹𝐵) = (𝐺𝐵) ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
 
Theoremfnmptfvd 6360* A function with a given domain is a mapping defined by its function values. (Contributed by AV, 1-Mar-2019.)
(𝜑𝑀 Fn 𝐴)    &   (𝑖 = 𝑎𝐷 = 𝐶)    &   ((𝜑𝑖𝐴) → 𝐷𝑈)    &   ((𝜑𝑎𝐴) → 𝐶𝑉)       (𝜑 → (𝑀 = (𝑎𝐴𝐶) ↔ ∀𝑖𝐴 (𝑀𝑖) = 𝐷))
 
Theoremfndmdif 6361* Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) ≠ (𝐺𝑥)})
 
Theoremfndmdifcom 6362 The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = dom (𝐺𝐹))
 
Theoremfndmdifeq0 6363 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 𝐴𝐺 Fn 𝐴) → (dom (𝐹𝐺) = ∅ ↔ 𝐹 = 𝐺))
 
Theoremfndmin 6364* Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → dom (𝐹𝐺) = {𝑥𝐴 ∣ (𝐹𝑥) = (𝐺𝑥)})
 
Theoremfneqeql 6365 Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ dom (𝐹𝐺) = 𝐴))
 
Theoremfneqeql2 6366 Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺𝐴 ⊆ dom (𝐹𝐺)))
 
Theoremfnreseql 6367 Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.)
((𝐹 Fn 𝐴𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = (𝐺𝑋) ↔ 𝑋 ⊆ dom (𝐹𝐺)))
 
Theoremchfnrn 6368* 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 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → ran 𝐹 𝐴)
 
Theoremfunfvop 6369 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹𝐴)⟩ ∈ 𝐹)
 
Theoremfunfvbrb 6370 Two ways to say that 𝐴 is in the domain of 𝐹. (Contributed by Mario Carneiro, 1-May-2014.)
(Fun 𝐹 → (𝐴 ∈ dom 𝐹𝐴𝐹(𝐹𝐴)))
 
Theoremfvimacnvi 6371 A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.)
((Fun 𝐹𝐴 ∈ (𝐹𝐵)) → (𝐹𝐴) ∈ 𝐵)
 
Theoremfvimacnv 6372 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 6010 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
 
Theoremfunimass3 6373 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 6372 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 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
 
Theoremfunimass5 6374* A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐴 ⊆ (𝐹𝐵) ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremfunconstss 6375* Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.)
((Fun 𝐹𝐴 ⊆ dom 𝐹) → (∀𝑥𝐴 (𝐹𝑥) = 𝐵𝐴 ⊆ (𝐹 “ {𝐵})))
 
TheoremfvimacnvALT 6376 Alternate proof of fvimacnv 6372, based on funimass3 6373. If funimass3 6373 is ever proved directly, as opposed to using funimacnv 6008 pointwise, then the proof of funimacnv 6008 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → ((𝐹𝐴) ∈ 𝐵𝐴 ∈ (𝐹𝐵)))
 
Theoremelpreima 6377 Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝐹 Fn 𝐴 → (𝐵 ∈ (𝐹𝐶) ↔ (𝐵𝐴 ∧ (𝐹𝐵) ∈ 𝐶)))
 
Theoremfniniseg 6378 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}) ↔ (𝐶𝐴 ∧ (𝐹𝐶) = 𝐵)))
 
Theoremfncnvima2 6379* Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐹 Fn 𝐴 → (𝐹𝐵) = {𝑥𝐴 ∣ (𝐹𝑥) ∈ 𝐵})
 
Theoremfniniseg2 6380* Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
(𝐹 Fn 𝐴 → (𝐹 “ {𝐵}) = {𝑥𝐴 ∣ (𝐹𝑥) = 𝐵})
 
Theoremunpreima 6381 Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∪ (𝐹𝐵)))
 
Theoreminpreima 6382 Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∩ (𝐹𝐵)))
 
Theoremdifpreima 6383 Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.)
(Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
 
Theoremrespreima 6384 The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)
(Fun 𝐹 → ((𝐹𝐵) “ 𝐴) = ((𝐹𝐴) ∩ 𝐵))
 
Theoremiinpreima 6385* Preimage of an intersection. (Contributed by FL, 16-Apr-2012.)
((Fun 𝐹𝐴 ≠ ∅) → (𝐹 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐹𝐵))
 
Theoremintpreima 6386* Preimage of an intersection. (Contributed by FL, 28-Apr-2012.)
((Fun 𝐹𝐴 ≠ ∅) → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))
 
Theoremfimacnv 6387 The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)
(𝐹:𝐴𝐵 → (𝐹𝐵) = 𝐴)
 
Theoremfimacnvinrn 6388 Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 21-Jan-2017.)
(Fun 𝐹 → (𝐹𝐴) = (𝐹 “ (𝐴 ∩ ran 𝐹)))
 
Theoremfimacnvinrn2 6389 Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 17-Feb-2017.)
((Fun 𝐹 ∧ ran 𝐹𝐵) → (𝐹𝐴) = (𝐹 “ (𝐴𝐵)))
 
Theoremfvn0ssdmfun 6390* If a class' function values for certain arguments is not the empty set, the arguments are contained in the domain of the class, and the class restricted to the arguments is a function, analogous to fvfundmfvn0 6264. (Contributed by AV, 27-Jan-2020.)
(∀𝑎𝐷 (𝐹𝑎) ≠ ∅ → (𝐷 ⊆ dom 𝐹 ∧ Fun (𝐹𝐷)))
 
Theoremfnopfv 6391 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.)
((𝐹 Fn 𝐴𝐵𝐴) → ⟨𝐵, (𝐹𝐵)⟩ ∈ 𝐹)
 
Theoremfvelrn 6392 A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)
 
Theoremnelrnfvne 6393 A function value cannot be any element not contained in the range of the function. (Contributed by AV, 28-Jan-2020.)
((Fun 𝐹𝑋 ∈ dom 𝐹𝑌 ∉ ran 𝐹) → (𝐹𝑋) ≠ 𝑌)
 
Theoremfveqdmss 6394* If the empty set is not contained in the range of a function, and the function values of another class (not necessarily a function) are equal to the function values of the function for all elements of the domain of the function, then the domain of the function is contained in the domain of the class. (Contributed by AV, 28-Jan-2020.)
𝐷 = dom 𝐵       ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → 𝐷 ⊆ dom 𝐴)
 
Theoremfveqressseq 6395* If the empty set is not contained in the range of a function, and the function values of another class (not necessarily a function) are equal to the function values of the function for all elements of the domain of the function, then the class restricted to the domain of the function is the function itself. (Contributed by AV, 28-Jan-2020.)
𝐷 = dom 𝐵       ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥𝐷 (𝐴𝑥) = (𝐵𝑥)) → (𝐴𝐷) = 𝐵)
 
Theoremfnfvelrn 6396 A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)
((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) ∈ ran 𝐹)
 
Theoremffvelrn 6397 A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.)
((𝐹:𝐴𝐵𝐶𝐴) → (𝐹𝐶) ∈ 𝐵)
 
Theoremffvelrni 6398 A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.)
𝐹:𝐴𝐵       (𝐶𝐴 → (𝐹𝐶) ∈ 𝐵)
 
Theoremffvelrnda 6399 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐹:𝐴𝐵)       ((𝜑𝐶𝐴) → (𝐹𝐶) ∈ 𝐵)
 
Theoremffvelrnd 6400 A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐶𝐴)       (𝜑 → (𝐹𝐶) ∈ 𝐵)
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