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Theorem List for Metamath Proof Explorer - 30301-30400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremf1o3d 30301* Describe an implicit one-to-one onto function. (Contributed by Thierry Arnoux, 23-Apr-2017.)
(𝜑𝐹 = (𝑥𝐴𝐶))    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑦𝐵) → 𝐷𝐴)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))       (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))
 
Theoremeldmne0 30302 A function of nonempty domain is not empty. (Contributed by Thierry Arnoux, 20-Nov-2023.)
(𝑋 ∈ dom 𝐹𝐹 ≠ ∅)
 
Theoremf1rnen 30303 Equinumerosity of the range of an injective function. (Contributed by Thierry Arnoux, 7-Jul-2023.)
((𝐹:𝐴1-1𝐵𝐴𝑉) → ran 𝐹𝐴)
 
Theoremrinvf1o 30304 Sufficient conditions for the restriction of an involution to be a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
Fun 𝐹    &   𝐹 = 𝐹    &   (𝐹𝐴) ⊆ 𝐵    &   (𝐹𝐵) ⊆ 𝐴    &   𝐴 ⊆ dom 𝐹    &   𝐵 ⊆ dom 𝐹       (𝐹𝐴):𝐴1-1-onto𝐵
 
Theoremfresf1o 30305 Conditions for a restriction to be a one-to-one onto function. (Contributed by Thierry Arnoux, 7-Dec-2016.)
((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1-onto𝐶)
 
Theoremnfpconfp 30306 The set of fixed points of 𝐹 is the complement of the set of points moved by 𝐹. (Contributed by Thierry Arnoux, 17-Nov-2023.)
(𝐹 Fn 𝐴 → (𝐴 ∖ dom (𝐹 ∖ I )) = dom (𝐹 ∩ I ))
 
Theoremfmptco1f1o 30307* The action of composing (to the right) with a bijection is itself a bijection of functions. (Contributed by Thierry Arnoux, 3-Jan-2021.)
𝐴 = (𝑅m 𝐸)    &   𝐵 = (𝑅m 𝐷)    &   𝐹 = (𝑓𝐴 ↦ (𝑓𝑇))    &   (𝜑𝐷𝑉)    &   (𝜑𝐸𝑊)    &   (𝜑𝑅𝑋)    &   (𝜑𝑇:𝐷1-1-onto𝐸)       (𝜑𝐹:𝐴1-1-onto𝐵)
 
Theoremcofmpt2 30308* Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 15-Jul-2023.)
((𝜑𝑦 = (𝐹𝑥)) → 𝐶 = 𝐷)    &   ((𝜑𝑦𝐵) → 𝐶𝐸)    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐷𝑉)       (𝜑 → ((𝑦𝐵𝐶) ∘ 𝐹) = (𝑥𝐴𝐷))
 
Theoremf1mptrn 30309* Express injection for a mapping operation. (Contributed by Thierry Arnoux, 3-May-2020.)
((𝜑𝑥𝐴) → 𝐵𝐶)    &   ((𝜑𝑦𝐶) → ∃!𝑥𝐴 𝑦 = 𝐵)       (𝜑 → Fun (𝑥𝐴𝐵))
 
Theoremdfimafnf 30310* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.)
𝑥𝐴    &   𝑥𝐹       ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})
 
Theoremfunimass4f 30311 Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝐹       ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 
Theoremelimampt 30312* Membership in the image of a mapping. (Contributed by Thierry Arnoux, 3-Jan-2022.)
𝐹 = (𝑥𝐴𝐵)    &   (𝜑𝐶𝑊)    &   (𝜑𝐷𝐴)       (𝜑 → (𝐶 ∈ (𝐹𝐷) ↔ ∃𝑥𝐷 𝐶 = 𝐵))
 
Theoremsuppss2f 30313* Show that the support of a function is contained in a set. (Contributed by Thierry Arnoux, 22-Jun-2017.) (Revised by AV, 1-Sep-2020.)
𝑘𝜑    &   𝑘𝐴    &   𝑘𝑊    &   ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍)    &   (𝜑𝐴𝑉)       (𝜑 → ((𝑘𝐴𝐵) supp 𝑍) ⊆ 𝑊)
 
Theoremfovcld 30314 Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Revised by Thierry Arnoux, 17-Feb-2017.)
(𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)       ((𝜑𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)
 
Theoremofrn 30315 The range of the function operation. (Contributed by Thierry Arnoux, 8-Jan-2017.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑+ :(𝐵 × 𝐵)⟶𝐶)    &   (𝜑𝐴𝑉)       (𝜑 → ran (𝐹f + 𝐺) ⊆ 𝐶)
 
Theoremofrn2 30316 The range of the function operation. (Contributed by Thierry Arnoux, 21-Mar-2017.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑+ :(𝐵 × 𝐵)⟶𝐶)    &   (𝜑𝐴𝑉)       (𝜑 → ran (𝐹f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
 
Theoremoff2 30317* The function operation produces a function - alternative form with all antecedents as deduction. (Contributed by Thierry Arnoux, 17-Feb-2017.)
((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐵𝑇)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑 → (𝐴𝐵) = 𝐶)       (𝜑 → (𝐹f 𝑅𝐺):𝐶𝑈)
 
Theoremofresid 30318 Applying an operation restricted to the range of the functions does not change the function operation. (Contributed by Thierry Arnoux, 14-Feb-2018.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑𝐴𝑉)       (𝜑 → (𝐹f 𝑅𝐺) = (𝐹f (𝑅 ↾ (𝐵 × 𝐵))𝐺))
 
Theoremfimarab 30319* Expressing the image of a set as a restricted abstract builder. (Contributed by Thierry Arnoux, 27-Jan-2020.)
((𝐹:𝐴𝐵𝑋𝐴) → (𝐹𝑋) = {𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦})
 
Theoremunipreima 30320* Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.)
(Fun 𝐹 → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))
 
Theoremsspreima 30321 The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.)
((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))
 
Theoremopfv 30322 Value of a function producing ordered pairs. (Contributed by Thierry Arnoux, 3-Jan-2017.)
(((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) = ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩)
 
Theoremxppreima 30323 The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 6-Jun-2017.)
((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (𝐹 “ (𝑌 × 𝑍)) = (((1st𝐹) “ 𝑌) ∩ ((2nd𝐹) “ 𝑍)))
 
Theoremxppreima2 30324* The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 7-Jun-2017.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐶)    &   𝐻 = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)       (𝜑 → (𝐻 “ (𝑌 × 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))
 
Theoremelunirn2 30325 Condition for the membership in the union of the range of a function. (Contributed by Thierry Arnoux, 13-Nov-2016.)
((Fun 𝐹𝐵 ∈ (𝐹𝐴)) → 𝐵 ran 𝐹)
 
Theoremabfmpunirn 30326* Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 28-Sep-2016.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})    &   {𝑦𝜑} ∈ V    &   (𝑦 = 𝐵 → (𝜑𝜓))       (𝐵 ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 𝜓))
 
Theoremrabfmpunirn 30327* Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 30-Sep-2016.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝑊𝜑})    &   𝑊 ∈ V    &   (𝑦 = 𝐵 → (𝜑𝜓))       (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 (𝐵𝑊𝜓))
 
Theoremabfmpeld 30328* Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝜓})    &   (𝜑 → {𝑦𝜓} ∈ V)    &   (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)))       (𝜑 → ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜒)))
 
Theoremabfmpel 30329* Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})    &   {𝑦𝜑} ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜓))
 
TheoremfmptdF 30330 Domain and codomain of the mapping operation; deduction form. This version of fmptd 6871 uses bound-variable hypothesis instead of distinct variable conditions. (Contributed by Thierry Arnoux, 28-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐶    &   ((𝜑𝑥𝐴) → 𝐵𝐶)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑𝐹:𝐴𝐶)
 
Theoremfmptcof2 30331* Composition of two functions expressed as ordered-pair class abstractions. (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.) (Revised by Thierry Arnoux, 10-May-2017.)
𝑥𝑆    &   𝑦𝑇    &   𝑥𝐴    &   𝑥𝐵    &   𝑥𝜑    &   (𝜑 → ∀𝑥𝐴 𝑅𝐵)    &   (𝜑𝐹 = (𝑥𝐴𝑅))    &   (𝜑𝐺 = (𝑦𝐵𝑆))    &   (𝑦 = 𝑅𝑆 = 𝑇)       (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))
 
Theoremfcomptf 30332* Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 6888. (Contributed by Thierry Arnoux, 30-Jun-2017.)
𝑥𝐵       ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))
 
Theoremacunirnmpt 30333* Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 6-Nov-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)    &   𝐶 = ran (𝑗𝐴𝐵)       (𝜑 → ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶𝑗𝐴 (𝑓𝑦) ∈ 𝐵))
 
Theoremacunirnmpt2 30334* Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)    &   𝐶 = ran (𝑗𝐴𝐵)    &   (𝑗 = (𝑓𝑥) → 𝐵 = 𝐷)       (𝜑 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))
 
Theoremacunirnmpt2f 30335* Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)    &   𝑗𝐴    &   𝑗𝐶    &   𝑗𝐷    &   𝐶 = 𝑗𝐴 𝐵    &   (𝑗 = (𝑓𝑥) → 𝐵 = 𝐷)    &   ((𝜑𝑗𝐴) → 𝐵𝑊)       (𝜑 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))
 
Theoremaciunf1lem 30336* Choice in an index union. (Contributed by Thierry Arnoux, 8-Nov-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)    &   𝑗𝐴    &   ((𝜑𝑗𝐴) → 𝐵𝑊)       (𝜑 → ∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 𝑗𝐴 𝐵(2nd ‘(𝑓𝑥)) = 𝑥))
 
Theoremaciunf1 30337* Choice in an index union. (Contributed by Thierry Arnoux, 4-May-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵𝑊)       (𝜑 → ∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑘 𝑗𝐴 𝐵(2nd ‘(𝑓𝑘)) = 𝑘))
 
Theoremofoprabco 30338* Function operation as a composition with an operation. (Contributed by Thierry Arnoux, 4-Jun-2017.)
𝑎𝑀    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝑀 = (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩))    &   (𝜑𝑁 = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)))       (𝜑 → (𝐹f 𝑅𝐺) = (𝑁𝑀))
 
Theoremofpreima 30339* Express the preimage of a function operation as a union of preimages. (Contributed by Thierry Arnoux, 8-Mar-2018.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝑅 Fn (𝐵 × 𝐶))       (𝜑 → ((𝐹f 𝑅𝐺) “ 𝐷) = 𝑝 ∈ (𝑅𝐷)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
 
Theoremofpreima2 30340* Express the preimage of a function operation as a union of preimages. This version of ofpreima 30339 iterates the union over a smaller set. (Contributed by Thierry Arnoux, 8-Mar-2018.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝑅 Fn (𝐵 × 𝐶))       (𝜑 → ((𝐹f 𝑅𝐺) “ 𝐷) = 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
 
Theoremfuncnvmpt 30341* Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐹    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (Fun 𝐹 ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))
 
Theoremfuncnv5mpt 30342* Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 1-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐹    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝑥 = 𝑧𝐵 = 𝐶)       (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝐴𝑧𝐴 (𝑥 = 𝑧𝐵𝐶)))
 
Theoremfuncnv4mpt 30343* Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 2-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐹    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (Fun 𝐹 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)))
 
Theorempreimane 30344 Different elements have different preimages. (Contributed by Thierry Arnoux, 7-May-2023.)
(𝜑 → Fun 𝐹)    &   (𝜑𝑋𝑌)    &   (𝜑𝑋 ∈ ran 𝐹)    &   (𝜑𝑌 ∈ ran 𝐹)       (𝜑 → (𝐹 “ {𝑋}) ≠ (𝐹 “ {𝑌}))
 
Theoremfnpreimac 30345* Choose a set 𝑥 containing a preimage of each element of a given set 𝐵. (Contributed by Thierry Arnoux, 7-May-2023.)
((𝐴𝑉𝐹 Fn 𝐴𝐵 ⊆ ran 𝐹) → ∃𝑥 ∈ 𝒫 𝐴(𝑥𝐵 ∧ (𝐹𝑥) = 𝐵))
 
Theoremfgreu 30346* Exactly one point of a function's graph has a given first element. (Contributed by Thierry Arnoux, 1-Apr-2018.)
((Fun 𝐹𝑋 ∈ dom 𝐹) → ∃!𝑝𝐹 𝑋 = (1st𝑝))
 
Theoremfcnvgreu 30347* If the converse of a relation 𝐴 is a function, exactly one point of its graph has a given second element (that is, function value). (Contributed by Thierry Arnoux, 1-Apr-2018.)
(((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑝𝐴 𝑌 = (2nd𝑝))
 
Theoremrnmposs 30348* The range of an operation given by the maps-to notation as a subset. (Contributed by Thierry Arnoux, 23-May-2017.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → ran 𝐹𝐷)
 
TheoremmptssALT 30349* Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 5904. (Contributed by Thierry Arnoux, 30-May-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))
 
Theorempartfun 30350 Rewrite a function defined by parts, using a mapping and an if construct, into a union of functions on disjoint domains. (Contributed by Thierry Arnoux, 30-Mar-2017.)
(𝑥𝐴 ↦ if(𝑥𝐵, 𝐶, 𝐷)) = ((𝑥 ∈ (𝐴𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴𝐵) ↦ 𝐷))
 
Theoremdfcnv2 30351* Alternative definition of the converse of a relation. (Contributed by Thierry Arnoux, 31-Mar-2018.)
(ran 𝑅𝐴𝑅 = 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})))
 
Theoremfnimatp 30352 The image of a triplet under a function. (Contributed by Thierry Arnoux, 19-Sep-2023.)
(𝜑𝐹 Fn 𝐷)    &   (𝜑𝐴𝐷)    &   (𝜑𝐵𝐷)    &   (𝜑𝐶𝐷)       (𝜑 → (𝐹 “ {𝐴, 𝐵, 𝐶}) = {(𝐹𝐴), (𝐹𝐵), (𝐹𝐶)})
 
Theoremfnunres2 30353 Restriction of a disjoint union to the domain of the second function. (Contributed by Thierry Arnoux, 12-Oct-2023.)
((𝐹 Fn 𝐴𝐺 Fn 𝐵 ∧ (𝐴𝐵) = ∅) → ((𝐹𝐺) ↾ 𝐵) = 𝐺)
 
20.3.4.3  Operations - misc additions
 
Theoremmpomptxf 30354* Express a two-argument function as a one-argument function, or vice-versa. In this version 𝐵(𝑥) is not assumed to be constant w.r.t 𝑥. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Thierry Arnoux, 31-Mar-2018.)
𝑥𝐶    &   𝑦𝐶    &   (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)       (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
 
Theoremsuppovss 30355* A bound for the support of an operation. (Contributed by Thierry Arnoux, 19-Jul-2023.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   𝐺 = (𝑥𝐴 ↦ (𝑦𝐵𝐶))    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝑍𝐷)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝐷)       (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐺 supp (𝐵 × {𝑍})) × 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺𝑘) supp 𝑍)))
 
20.3.4.4  Explicit Functions with one or two points as a domain
 
Theorembrsnop 30356 Binary relation for an ordered pair singleton. (Contributed by Thierry Arnoux, 23-Sep-2023.)
((𝐴𝑉𝐵𝑊) → (𝑋{⟨𝐴, 𝐵⟩}𝑌 ↔ (𝑋 = 𝐴𝑌 = 𝐵)))
 
Theoremcosnopne 30357 Composition of two ordered pair singletons with non-matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)    &   (𝜑𝐴𝐷)       (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐷⟩}) = ∅)
 
Theoremcosnop 30358 Composition of two ordered pair singletons with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑋)       (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐴⟩}) = {⟨𝐶, 𝐵⟩})
 
Theoremcnvprop 30359 Converse of a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(((𝐴𝑉𝐵𝑊) ∧ (𝐶𝑉𝐷𝑊)) → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {⟨𝐵, 𝐴⟩, ⟨𝐷, 𝐶⟩})
 
Theorembrprop 30360 Binary relation for a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)       (𝜑 → (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌 ↔ ((𝑋 = 𝐴𝑌 = 𝐵) ∨ (𝑋 = 𝐶𝑌 = 𝐷))))
 
Theoremmptprop 30361* Rewrite pairs of ordered pairs as mapping to functions. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)    &   (𝜑𝐴𝐶)       (𝜑 → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
 
Theoremcoprprop 30362 Composition of two pairs of ordered pairs with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑊)    &   (𝜑𝐴𝐶)    &   (𝜑𝐸𝑋)    &   (𝜑𝐹𝑋)    &   (𝜑𝐸𝐹)       (𝜑 → ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∘ {⟨𝐸, 𝐴⟩, ⟨𝐹, 𝐶⟩}) = {⟨𝐸, 𝐵⟩, ⟨𝐹, 𝐷⟩})
 
20.3.4.5  Isomorphisms - misc. add.
 
Theoremgtiso 30363 Two ways to write a strictly decreasing function on the reals. (Contributed by Thierry Arnoux, 6-Apr-2017.)
((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
 
Theoremisoun 30364* Infer an isomorphism from a union of two isomorphisms. (Contributed by Thierry Arnoux, 30-Mar-2017.)
(𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷))    &   ((𝜑𝑥𝐴𝑦𝐶) → 𝑥𝑅𝑦)    &   ((𝜑𝑧𝐵𝑤𝐷) → 𝑧𝑆𝑤)    &   ((𝜑𝑥𝐶𝑦𝐴) → ¬ 𝑥𝑅𝑦)    &   ((𝜑𝑧𝐷𝑤𝐵) → ¬ 𝑧𝑆𝑤)    &   (𝜑 → (𝐴𝐶) = ∅)    &   (𝜑 → (𝐵𝐷) = ∅)       (𝜑 → (𝐻𝐺) Isom 𝑅, 𝑆 ((𝐴𝐶), (𝐵𝐷)))
 
20.3.4.6  Disjointness (additional proof requiring functions)
 
Theoremdisjdsct 30365* A disjoint collection is distinct, i.e. each set in this collection is different of all others, provided that it does not contain the empty set This can be expressed as "the converse of the mapping function is a function", or "the mapping function is single-rooted". (Cf. funcnv 6417) (Contributed by Thierry Arnoux, 28-Feb-2017.)
𝑥𝜑    &   𝑥𝐴    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (𝑉 ∖ {∅}))    &   (𝜑Disj 𝑥𝐴 𝐵)       (𝜑 → Fun (𝑥𝐴𝐵))
 
20.3.4.7  First and second members of an ordered pair - misc additions
 
Theoremdf1stres 30366* Definition for a restriction of the 1st (first member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
(1st ↾ (𝐴 × 𝐵)) = (𝑥𝐴, 𝑦𝐵𝑥)
 
Theoremdf2ndres 30367* Definition for a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
(2nd ↾ (𝐴 × 𝐵)) = (𝑥𝐴, 𝑦𝐵𝑦)
 
Theorem1stpreimas 30368 The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.)
((Rel 𝐴𝑋𝑉) → ((1st𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋})))
 
Theorem1stpreima 30369 The preimage by 1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
(𝐴𝐵 → ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))
 
Theorem2ndpreima 30370 The preimage by 2nd is an 'horizontal band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
(𝐴𝐶 → ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))
 
Theoremcurry2ima 30371* The image of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 25-Sep-2017.)
𝐺 = (𝐹(1st ↾ (V × {𝐶})))       ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (𝐺𝐷) = {𝑦 ∣ ∃𝑥𝐷 𝑦 = (𝑥𝐹𝐶)})
 
20.3.4.8  Supremum - misc additions
 
Theoremsupssd 30372* Inequality deduction for supremum of a subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))       (𝜑 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅))
 
Theoreminfssd 30373* Inequality deduction for infimum of a subset. (Contributed by AV, 4-Oct-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐶𝐵)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐶 𝑧𝑅𝑦)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ¬ inf(𝐶, 𝐴, 𝑅)𝑅inf(𝐵, 𝐴, 𝑅))
 
20.3.4.9  Finite Sets
 
Theoremimafi2 30374 The image by a finite set is finite. See also imafi 8806. (Contributed by Thierry Arnoux, 25-Apr-2020.)
(𝐴 ∈ Fin → (𝐴𝐵) ∈ Fin)
 
Theoremunifi3 30375 If a union is finite, then all its elements are finite. See unifi 8802. (Contributed by Thierry Arnoux, 27-Aug-2017.)
( 𝐴 ∈ Fin → 𝐴 ⊆ Fin)
 
20.3.4.10  Countable Sets
 
Theoremsnct 30376 A singleton is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.)
(𝐴𝑉 → {𝐴} ≼ ω)
 
Theoremprct 30377 An unordered pair is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ≼ ω)
 
Theoremmpocti 30378* An operation is countable if both its domains are countable. (Contributed by Thierry Arnoux, 17-Sep-2017.)
𝑥𝐴𝑦𝐵 𝐶𝑉       ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝑥𝐴, 𝑦𝐵𝐶) ≼ ω)
 
Theoremabrexct 30379* An image set of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≼ ω)
 
Theoremmptctf 30380 A countable mapping set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝐴       (𝐴 ≼ ω → (𝑥𝐴𝐵) ≼ ω)
 
Theoremabrexctf 30381* An image set of a countable set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝐴       (𝐴 ≼ ω → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≼ ω)
 
Theorempadct 30382* Index a countable set with integers and pad with 𝑍. (Contributed by Thierry Arnoux, 1-Jun-2020.)
((𝐴 ≼ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))
 
TheoremcnvoprabOLD 30383* The converse of a class abstraction of nested ordered pairs. Obsolete version of cnvoprab 7749 as of 16-Oct-2022, which has nonfreeness hypotheses instead of disjoint variable conditions. (Contributed by Thierry Arnoux, 17-Aug-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜓    &   𝑦𝜓    &   (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓𝜑))    &   (𝜓𝑎 ∈ (V × V))       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
 
Theoremf1od2 30384* Sufficient condition for a binary function expressed in maps-to notation to be bijective. (Contributed by Thierry Arnoux, 17-Aug-2017.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝑊)    &   ((𝜑𝑧𝐷) → (𝐼𝑋𝐽𝑌))    &   (𝜑 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽))))       (𝜑𝐹:(𝐴 × 𝐵)–1-1-onto𝐷)
 
Theoremfcobij 30385* Composing functions with a bijection yields a bijection between sets of functions. (Contributed by Thierry Arnoux, 25-Aug-2017.)
(𝜑𝐺:𝑆1-1-onto𝑇)    &   (𝜑𝑅𝑈)    &   (𝜑𝑆𝑉)    &   (𝜑𝑇𝑊)       (𝜑 → (𝑓 ∈ (𝑆m 𝑅) ↦ (𝐺𝑓)):(𝑆m 𝑅)–1-1-onto→(𝑇m 𝑅))
 
Theoremfcobijfs 30386* Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfien 8860. (Contributed by Thierry Arnoux, 25-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
(𝜑𝐺:𝑆1-1-onto𝑇)    &   (𝜑𝑅𝑈)    &   (𝜑𝑆𝑉)    &   (𝜑𝑇𝑊)    &   (𝜑𝑂𝑆)    &   𝑄 = (𝐺𝑂)    &   𝑋 = {𝑔 ∈ (𝑆m 𝑅) ∣ 𝑔 finSupp 𝑂}    &   𝑌 = { ∈ (𝑇m 𝑅) ∣ finSupp 𝑄}       (𝜑 → (𝑓𝑋 ↦ (𝐺𝑓)):𝑋1-1-onto𝑌)
 
Theoremsuppss3 30387* Deduce a function's support's inclusion in another function's support. (Contributed by Thierry Arnoux, 7-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝐺 = (𝑥𝐴𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝑍𝑊)    &   (𝜑𝐹 Fn 𝐴)    &   ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑍) → 𝐵 = 𝑍)       (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))
 
Theoremfsuppcurry1 30388* Finite support of a curried function with a constant first argument. (Contributed by Thierry Arnoux, 7-Jul-2023.)
𝐺 = (𝑥𝐵 ↦ (𝐶𝐹𝑥))    &   (𝜑𝑍𝑈)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐹 Fn (𝐴 × 𝐵))    &   (𝜑𝐶𝐴)    &   (𝜑𝐹 finSupp 𝑍)       (𝜑𝐺 finSupp 𝑍)
 
Theoremfsuppcurry2 30389* Finite support of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 7-Jul-2023.)
𝐺 = (𝑥𝐴 ↦ (𝑥𝐹𝐶))    &   (𝜑𝑍𝑈)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐹 Fn (𝐴 × 𝐵))    &   (𝜑𝐶𝐵)    &   (𝜑𝐹 finSupp 𝑍)       (𝜑𝐺 finSupp 𝑍)
 
Theoremoffinsupp1 30390* Finite support for a function operation. (Contributed by Thierry Arnoux, 8-Jul-2023.)
(𝜑𝐴𝑉)    &   (𝜑𝑌𝑈)    &   (𝜑𝑍𝑊)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐴𝑇)    &   (𝜑𝐹 finSupp 𝑌)    &   ((𝜑𝑥𝑇) → (𝑌𝑅𝑥) = 𝑍)       (𝜑 → (𝐹f 𝑅𝐺) finSupp 𝑍)
 
Theoremffs2 30391 Rewrite a function's support based with its range rather than the universal class. See also frnsuppeq 7833. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝐶 = (𝐵 ∖ {𝑍})       ((𝐴𝑉𝑍𝑊𝐹:𝐴𝐵) → (𝐹 supp 𝑍) = (𝐹𝐶))
 
Theoremffsrn 30392 The range of a finitely supported function is finite. (Contributed by Thierry Arnoux, 27-Aug-2017.)
(𝜑𝑍𝑊)    &   (𝜑𝐹𝑉)    &   (𝜑 → Fun 𝐹)    &   (𝜑 → (𝐹 supp 𝑍) ∈ Fin)       (𝜑 → ran 𝐹 ∈ Fin)
 
Theoremresf1o 30393* Restriction of functions to a superset of their support creates a bijection. (Contributed by Thierry Arnoux, 12-Sep-2017.)
𝑋 = {𝑓 ∈ (𝐵m 𝐴) ∣ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶}    &   𝐹 = (𝑓𝑋 ↦ (𝑓𝐶))       (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → 𝐹:𝑋1-1-onto→(𝐵m 𝐶))
 
Theoremmaprnin 30394* Restricting the range of the mapping operator. (Contributed by Thierry Arnoux, 30-Aug-2017.)
𝐴 ∈ V    &   𝐵 ∈ V       ((𝐵𝐶) ↑m 𝐴) = {𝑓 ∈ (𝐵m 𝐴) ∣ ran 𝑓𝐶}
 
Theoremfpwrelmapffslem 30395* Lemma for fpwrelmapffs 30397. For this theorem, the sets 𝐴 and 𝐵 could be infinite, but the relation 𝑅 itself is finite. (Contributed by Thierry Arnoux, 1-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝜑𝐹:𝐴⟶𝒫 𝐵)    &   (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))})       (𝜑 → (𝑅 ∈ Fin ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))
 
Theoremfpwrelmap 30396* Define a canonical mapping between functions from 𝐴 into subsets of 𝐵 and the relations with domain 𝐴 and range within 𝐵. Note that the same relation is used in axdc2lem 9859 and marypha2lem1 8888. (Contributed by Thierry Arnoux, 28-Aug-2017.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝑀 = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})       𝑀:(𝒫 𝐵m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)
 
Theoremfpwrelmapffs 30397* Define a canonical mapping between finite relations (finite subsets of a cartesian product) and functions with finite support into finite subsets. (Contributed by Thierry Arnoux, 28-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝑀 = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})    &   𝑆 = {𝑓 ∈ ((𝒫 𝐵 ∩ Fin) ↑m 𝐴) ∣ (𝑓 supp ∅) ∈ Fin}       (𝑀𝑆):𝑆1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin)
 
20.3.5  Real and Complex Numbers
 
20.3.5.1  Complex operations - misc. additions
 
Theoremcreq0 30398 The real representation of complex numbers is zero iff both its terms are zero. Cf. crne0 11620. (Contributed by Thierry Arnoux, 20-Aug-2023.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 = 0 ∧ 𝐵 = 0) ↔ (𝐴 + (i · 𝐵)) = 0))
 
Theorem1nei 30399 The imaginary unit i is not one. (Contributed by Thierry Arnoux, 20-Aug-2023.)
1 ≠ i
 
Theorem1neg1t1neg1 30400 An integer unit times itself. (Contributed by Thierry Arnoux, 23-Aug-2020.)
(𝑁 ∈ {-1, 1} → (𝑁 · 𝑁) = 1)
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