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Theorem List for Metamath Proof Explorer - 30001-30100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdfimafnf 30001* 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 30002 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 30003* 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 30004* 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 30005 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 30006 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 30007 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 30008* 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 30009 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 30010* 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 30011* Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.)
(Fun 𝐹 → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))

Theoremsspreima 30012 The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.)
((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))

Theoremopfv 30013 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 30014 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 30015* 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 30016 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 30017* 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 30018* 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 30019* 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 30020* 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 30021 Domain and codomain of the mapping operation; deduction form. This version of fmptd 6648 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 30022* 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 30023* Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 6665. (Contributed by Thierry Arnoux, 30-Jun-2017.)
𝑥𝐵       ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))

Theoremacunirnmpt 30024* 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 30025* 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 30026* 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 30027* 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 30028* 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 30029* 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 30030* 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 30031* Express the preimage of a function operation as a union of preimages. This version of ofpreima 30030 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 30032* 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 30033* 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 30034* 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 30035 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 30036* 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 30037* 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 30038* 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))

Theoremrnmpt2ss 30039* 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 30040* Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 5704. (Contributed by Thierry Arnoux, 30-May-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))

Theorempartfun 30041 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 30042* Alternative definition of the converse of a relation. (Contributed by Thierry Arnoux, 31-Mar-2018.)
(ran 𝑅𝐴𝑅 = 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})))

Theoremmpt2mptxf 30043* 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)

Theoremgtiso 30044 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 30045* 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)))

20.3.4.5  Disjointness (additional proof requiring functions)

Theoremdisjdsct 30046* 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 6203) (Contributed by Thierry Arnoux, 28-Feb-2017.)
𝑥𝜑    &   𝑥𝐴    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (𝑉 ∖ {∅}))    &   (𝜑Disj 𝑥𝐴 𝐵)       (𝜑 → Fun (𝑥𝐴𝐵))

20.3.4.6  First and second members of an ordered pair - misc additions

Theoremdf1stres 30047* 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 30048* 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 30049 The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.)
((Rel 𝐴𝑋𝑉) → ((1st𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋})))

Theorem1stpreima 30050 The preimage by 1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
(𝐴𝐵 → ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))

Theorem2ndpreima 30051 The preimage by 2nd is an 'horizontal band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
(𝐴𝐶 → ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))

Theoremcurry2ima 30052* 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)})

Theoremsupssd 30053* 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 30054* 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))

20.3.4.8  Finite Sets

Theoremimafi2 30055 The image by a finite set is finite. See also imafi 8547. (Contributed by Thierry Arnoux, 25-Apr-2020.)
(𝐴 ∈ Fin → (𝐴𝐵) ∈ Fin)

Theoremunifi3 30056 If a union is finite, then all its elements are finite. See unifi 8543. (Contributed by Thierry Arnoux, 27-Aug-2017.)
( 𝐴 ∈ Fin → 𝐴 ⊆ Fin)

20.3.4.9  Countable Sets

Theoremsnct 30057 A singleton is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.)
(𝐴𝑉 → {𝐴} ≼ ω)

Theoremprct 30058 An unordered pair is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ≼ ω)

Theoremmpt2cti 30059* 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 30060* An image set of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≼ ω)

Theoremmptctf 30061 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 30062* 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 30063* 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 30064* The converse of a class abstraction of nested ordered pairs. Obsolete version of cnvoprab 7509 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 30065* 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 30066* 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 30067* Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfien 8601. (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 30068* 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))

Theoremffs2 30069 Rewrite a function's support based with its range rather than the universal class. See also frnsuppeq 7588. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝐶 = (𝐵 ∖ {𝑍})       ((𝐴𝑉𝑍𝑊𝐹:𝐴𝐵) → (𝐹 supp 𝑍) = (𝐹𝐶))

Theoremffsrn 30070 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 30071* 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 30072* 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 30073* Lemma for fpwrelmapffs 30075. 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 30074* 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 9605 and marypha2lem1 8629. (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 30075* 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

Theoremsubeqxfrd 30076 Transfer two terms of a subtraction in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑 → (𝐴𝐵) = (𝐶𝐷))       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremznsqcld 30077 The square of a nonzero integer is a positive integer. (Contributed by Thierry Arnoux, 2-Feb-2020.)
(𝜑𝑁 ∈ ℤ)    &   (𝜑𝑁 ≠ 0)       (𝜑 → (𝑁↑2) ∈ ℕ)

Theoremnn0sqeq1 30078 A natural number with square one is equal to one. (Contributed by Thierry Arnoux, 2-Feb-2020.)
((𝑁 ∈ ℕ0 ∧ (𝑁↑2) = 1) → 𝑁 = 1)

Theorem1neg1t1neg1 30079 An integer unit times itself. (Contributed by Thierry Arnoux, 23-Aug-2020.)
(𝑁 ∈ {-1, 1} → (𝑁 · 𝑁) = 1)

Theoremnnmulge 30080 Multiplying by a positive integer 𝑀 yields greater than or equal nonnegative integers. (Contributed by Thierry Arnoux, 13-Dec-2021.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → 𝑁 ≤ (𝑀 · 𝑁))

20.3.5.2  Ordering on reals - misc additions

Theoremlt2addrd 30081* If the right-hand side of a 'less than' relationship is an addition, then we can express the left-hand side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 < (𝐵 + 𝐶))       (𝜑 → ∃𝑏 ∈ ℝ ∃𝑐 ∈ ℝ (𝐴 = (𝑏 + 𝑐) ∧ 𝑏 < 𝐵𝑐 < 𝐶))

20.3.5.3  Extended reals - misc additions

Theoremxrlelttric 30082 Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵𝐵 < 𝐴))

Theoremxaddeq0 30083 Two extended reals which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 13-Jun-2017.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → ((𝐴 +𝑒 𝐵) = 0 ↔ 𝐴 = -𝑒𝐵))

Theoremxrinfm 30084 The extended real numbers are unbounded below. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Revised by AV, 28-Sep-2020.)
inf(ℝ*, ℝ*, < ) = -∞

Theoremle2halvesd 30085 A sum is less than the whole if each term is less than half. (Contributed by Thierry Arnoux, 29-Nov-2017.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ)    &   (𝜑𝐶 ∈ ℝ)    &   (𝜑𝐴 ≤ (𝐶 / 2))    &   (𝜑𝐵 ≤ (𝐶 / 2))       (𝜑 → (𝐴 + 𝐵) ≤ 𝐶)

Theoremxraddge02 30086 A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 28-Dec-2016.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (0 ≤ 𝐵𝐴 ≤ (𝐴 +𝑒 𝐵)))

Theoremxrge0addge 30087 A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 19-Jul-2020.)
((𝐴 ∈ ℝ*𝐵 ∈ (0[,]+∞)) → 𝐴 ≤ (𝐴 +𝑒 𝐵))

Theoremxlt2addrd 30088* If the right-hand side of a 'less than' relationship is an addition, then we can express the left-hand side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)
(𝜑𝐴 ∈ ℝ)    &   (𝜑𝐵 ∈ ℝ*)    &   (𝜑𝐶 ∈ ℝ*)    &   (𝜑𝐵 ≠ -∞)    &   (𝜑𝐶 ≠ -∞)    &   (𝜑𝐴 < (𝐵 +𝑒 𝐶))       (𝜑 → ∃𝑏 ∈ ℝ*𝑐 ∈ ℝ* (𝐴 = (𝑏 +𝑒 𝑐) ∧ 𝑏 < 𝐵𝑐 < 𝐶))

Theoremxrsupssd 30089 Inequality deduction for supremum of an extended real subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)
(𝜑𝐵𝐶)    &   (𝜑𝐶 ⊆ ℝ*)       (𝜑 → sup(𝐵, ℝ*, < ) ≤ sup(𝐶, ℝ*, < ))

Theoremxrge0infss 30090* Any subset of nonnegative extended reals has an infimum. (Contributed by Thierry Arnoux, 16-Sep-2019.) (Revised by AV, 4-Oct-2020.)
(𝐴 ⊆ (0[,]+∞) → ∃𝑥 ∈ (0[,]+∞)(∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ (0[,]+∞)(𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))

Theoremxrge0infssd 30091 Inequality deduction for infimum of a nonnegative extended real subset. (Contributed by Thierry Arnoux, 16-Sep-2019.) (Revised by AV, 4-Oct-2020.)
(𝜑𝐶𝐵)    &   (𝜑𝐵 ⊆ (0[,]+∞))       (𝜑 → inf(𝐵, (0[,]+∞), < ) ≤ inf(𝐶, (0[,]+∞), < ))

Theoremxrge0addcld 30092 Nonnegative extended reals are closed under addition. (Contributed by Thierry Arnoux, 16-Sep-2019.)
(𝜑𝐴 ∈ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑 → (𝐴 +𝑒 𝐵) ∈ (0[,]+∞))

Theoremxrge0subcld 30093 Condition for closure of nonnegative extended reals under subtraction. (Contributed by Thierry Arnoux, 27-May-2020.)
(𝜑𝐴 ∈ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))    &   (𝜑𝐵𝐴)       (𝜑 → (𝐴 +𝑒 -𝑒𝐵) ∈ (0[,]+∞))

Theoreminfxrge0lb 30094 A member of a set of nonnegative extended reals is greater than or equal to the set's infimum. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.)
(𝜑𝐴 ⊆ (0[,]+∞))    &   (𝜑𝐵𝐴)       (𝜑 → inf(𝐴, (0[,]+∞), < ) ≤ 𝐵)

Theoreminfxrge0glb 30095* The infimum of a set of nonnegative extended reals is the greatest lower bound. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.)
(𝜑𝐴 ⊆ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑 → (inf(𝐴, (0[,]+∞), < ) < 𝐵 ↔ ∃𝑥𝐴 𝑥 < 𝐵))

Theoreminfxrge0gelb 30096* The infimum of a set of nonnegative extended reals is greater than or equal to a lower bound. (Contributed by Thierry Arnoux, 19-Jul-2020.) (Revised by AV, 4-Oct-2020.)
(𝜑𝐴 ⊆ (0[,]+∞))    &   (𝜑𝐵 ∈ (0[,]+∞))       (𝜑 → (𝐵 ≤ inf(𝐴, (0[,]+∞), < ) ↔ ∀𝑥𝐴 𝐵𝑥))

Theoremdfrp2 30097 Alternate definition of the positive real numbers. (Contributed by Thierry Arnoux, 4-May-2020.)
+ = (0(,)+∞)

Theoremxrofsup 30098 The supremum is preserved by extended addition set operation. (Provided minus infinity is not involved as it does not behave well with addition.) (Contributed by Thierry Arnoux, 20-Mar-2017.)
(𝜑𝑋 ⊆ ℝ*)    &   (𝜑𝑌 ⊆ ℝ*)    &   (𝜑 → sup(𝑋, ℝ*, < ) ≠ -∞)    &   (𝜑 → sup(𝑌, ℝ*, < ) ≠ -∞)    &   (𝜑𝑍 = ( +𝑒 “ (𝑋 × 𝑌)))       (𝜑 → sup(𝑍, ℝ*, < ) = (sup(𝑋, ℝ*, < ) +𝑒 sup(𝑌, ℝ*, < )))

Theoremsupxrnemnf 30099 The supremum of a nonempty set of extended reals which does not contain minus infinity is not minus infinity. (Contributed by Thierry Arnoux, 21-Mar-2017.)
((𝐴 ⊆ ℝ*𝐴 ≠ ∅ ∧ ¬ -∞ ∈ 𝐴) → sup(𝐴, ℝ*, < ) ≠ -∞)

20.3.5.4  Real number intervals - misc additions

Theoremjoiniooico 30100 Disjoint joining an open interval with a closed-below, open-above interval to form a closed-below, open-above interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
(((𝐴 ∈ ℝ*𝐵 ∈ ℝ*𝐶 ∈ ℝ*) ∧ (𝐴 < 𝐵𝐵𝐶)) → (((𝐴(,)𝐵) ∩ (𝐵[,)𝐶)) = ∅ ∧ ((𝐴(,)𝐵) ∪ (𝐵[,)𝐶)) = (𝐴(,)𝐶)))

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