P. 330 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32901-33000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	opfv 32901	Value of a function producing ordered pairs. (Contributed by Thierry Arnoux, 3-Jan-2017.)
⊢ (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) = ⟨((1st ∘ 𝐹)‘𝑥), ((2nd ∘ 𝐹)‘𝑥)⟩)
Theorem	xppreima 32902	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)) = ((◡(1st ∘ 𝐹) “ 𝑌) ∩ (◡(2nd ∘ 𝐹) “ 𝑍)))
Theorem	2ndimaxp 32903	Image of a cartesian product by 2nd. (Contributed by Thierry Arnoux, 23-Jun-2024.)
⊢ (𝐴 ≠ ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)
Theorem	dmdju 32904*	Domain of a disjoint union of non-empty sets. (Contributed by Thierry Arnoux, 5-Oct-2025.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≠ ∅)    ⇒   ⊢ (𝜑 → dom ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) = 𝐴)
Theorem	djussxp2 32905*	Stronger version of djussxp 5822. (Contributed by Thierry Arnoux, 23-Jun-2024.)
⊢ ∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵)
Theorem	2ndresdju 32906*	The 2nd function restricted to a disjoint union is injective. (Contributed by Thierry Arnoux, 23-Jun-2024.)
⊢ 𝑈 = ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑊)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝐶)    &   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴)    ⇒   ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1→𝐴)
Theorem	2ndresdjuf1o 32907*	The 2nd function restricted to a disjoint union is a bijection. See also e.g. 2ndconst 8084. (Contributed by Thierry Arnoux, 23-Jun-2024.)
⊢ 𝑈 = ∪ 𝑥 ∈ 𝑋 ({𝑥} × 𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑊)    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝑋 𝐶)    &   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝑋 𝐶 = 𝐴)    ⇒   ⊢ (𝜑 → (2nd ↾ 𝑈):𝑈–1-1-onto→𝐴)
Theorem	xppreima2 32908*	The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 7-Jun-2017.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐴 ↦ ⟨(𝐹‘𝑥), (𝐺‘𝑥)⟩)    ⇒   ⊢ (𝜑 → (◡𝐻 “ (𝑌 × 𝑍)) = ((◡𝐹 “ 𝑌) ∩ (◡𝐺 “ 𝑍)))
Theorem	abfmpunirn 32909*	Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 28-Sep-2016.)
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜑})    &   ⊢ {𝑦 ∣ 𝜑} ∈ V    &   ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥 ∈ 𝑉 𝜓))
Theorem	rabfmpunirn 32910*	Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 30-Sep-2016.)
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑊 ∣ 𝜑})    &   ⊢ 𝑊 ∈ V    &   ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓))
Theorem	abfmpeld 32911*	Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜓})    &   ⊢ (𝜑 → {𝑦 ∣ 𝜓} ∈ V)    &   ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜒)))
Theorem	abfmpel 32912*	Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜑})    &   ⊢ {𝑦 ∣ 𝜑} ∈ V    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∈ (𝐹‘𝐴) ↔ 𝜓))
Theorem	fmptdF 32913	Domain and codomain of the mapping operation; deduction form. This version of fmptd 7099 uses bound-variable hypothesis instead of distinct variable conditions. (Contributed by Thierry Arnoux, 28-Mar-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
Theorem	fmptcof2 32914*	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.)
⊢ Ⅎ𝑥𝑆    &   ⊢ Ⅎ𝑦𝑇    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝑅 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑅))    &   ⊢ (𝜑 → 𝐺 = (𝑦 ∈ 𝐵 ↦ 𝑆))    &   ⊢ (𝑦 = 𝑅 → 𝑆 = 𝑇)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝑇))
Theorem	fcomptf 32915*	Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 7119. (Contributed by Thierry Arnoux, 30-Jun-2017.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥))))
Theorem	acunirnmpt 32916*	Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 6-Nov-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅)    &   ⊢ 𝐶 = ran (𝑗 ∈ 𝐴 ↦ 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝐶⟶∪ 𝐶 ∧ ∀𝑦 ∈ 𝐶 ∃𝑗 ∈ 𝐴 (𝑓‘𝑦) ∈ 𝐵))
Theorem	acunirnmpt2 32917*	Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅)    &   ⊢ 𝐶 = ∪ ran (𝑗 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝑗 = (𝑓‘𝑥) → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷))
Theorem	acunirnmpt2f 32918*	Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅)    &   ⊢ Ⅎ𝑗𝐴    &   ⊢ Ⅎ𝑗𝐶    &   ⊢ Ⅎ𝑗𝐷    &   ⊢ 𝐶 = ∪ 𝑗 ∈ 𝐴 𝐵    &   ⊢ (𝑗 = (𝑓‘𝑥) → 𝐵 = 𝐷)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝐶⟶𝐴 ∧ ∀𝑥 ∈ 𝐶 𝑥 ∈ 𝐷))
Theorem	aciunf1lem 32919*	Choice in an index union. (Contributed by Thierry Arnoux, 8-Nov-2019.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ≠ ∅)    &   ⊢ Ⅎ𝑗𝐴    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑥)) = 𝑥))
Theorem	aciunf1 32920*	Choice in an index union. (Contributed by Thierry Arnoux, 4-May-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:∪ 𝑗 ∈ 𝐴 𝐵–1-1→∪ 𝑗 ∈ 𝐴 ({𝑗} × 𝐵) ∧ ∀𝑘 ∈ ∪ 𝑗 ∈ 𝐴 𝐵(2nd ‘(𝑓‘𝑘)) = 𝑘))
Theorem	ofoprabco 32921*	Function operation as a composition with an operation. (Contributed by Thierry Arnoux, 4-Jun-2017.)
⊢ Ⅎ𝑎𝑀    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑀 = (𝑎 ∈ 𝐴 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩))    &   ⊢ (𝜑 → 𝑁 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐶 ↦ (𝑥𝑅𝑦)))    ⇒   ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑁 ∘ 𝑀))
Theorem	ofpreima 32922*	Express the preimage of a function operation as a union of preimages. (Contributed by Thierry Arnoux, 8-Mar-2018.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 Fn (𝐵 × 𝐶))    ⇒   ⊢ (𝜑 → (◡(𝐹 ∘f 𝑅𝐺) “ 𝐷) = ∪ 𝑝 ∈ (◡𝑅 “ 𝐷)((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
Theorem	ofpreima2 32923*	Express the preimage of a function operation as a union of preimages. This version of ofpreima 32922 iterates the union over a smaller set. (Contributed by Thierry Arnoux, 8-Mar-2018.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 Fn (𝐵 × 𝐶))    ⇒   ⊢ (𝜑 → (◡(𝐹 ∘f 𝑅𝐺) “ 𝐷) = ∪ 𝑝 ∈ ((◡𝑅 “ 𝐷) ∩ (ran 𝐹 × ran 𝐺))((◡𝐹 “ {(1st ‘𝑝)}) ∩ (◡𝐺 “ {(2nd ‘𝑝)})))
Theorem	funcnv5mpt 32924*	Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 1-Mar-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    &   ⊢ (𝑥 = 𝑧 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥 = 𝑧 ∨ 𝐵 ≠ 𝐶)))
Theorem	funcnv4mpt 32925*	Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 2-Mar-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑖 ∈ 𝐴 ∀𝑗 ∈ 𝐴 (𝑖 = 𝑗 ∨ ⦋𝑖 / 𝑥⦌𝐵 ≠ ⦋𝑗 / 𝑥⦌𝐵)))
Theorem	preimane 32926	Different elements have different preimages. (Contributed by Thierry Arnoux, 7-May-2023.)
⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → 𝑋 ≠ 𝑌)    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐹)    &   ⊢ (𝜑 → 𝑌 ∈ ran 𝐹)    ⇒   ⊢ (𝜑 → (◡𝐹 “ {𝑋}) ≠ (◡𝐹 “ {𝑌}))
Theorem	fnpreimac 32927*	Choose a set 𝑥 containing a preimage of each element of a given set 𝐵. (Contributed by Thierry Arnoux, 7-May-2023.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ ran 𝐹) → ∃𝑥 ∈ 𝒫 𝐴(𝑥 ≈ 𝐵 ∧ (𝐹 “ 𝑥) = 𝐵))
Theorem	fgreu 32928*	Exactly one point of a function's graph has a given first element. (Contributed by Thierry Arnoux, 1-Apr-2018.)
⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∃!𝑝 ∈ 𝐹 𝑋 = (1st ‘𝑝))
Theorem	fcnvgreu 32929*	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) → ∃!𝑝 ∈ 𝐴 𝑌 = (2nd ‘𝑝))
Theorem	rnmposs 32930*	The range of an operation given by the maps-to notation as a subset. (Contributed by Thierry Arnoux, 23-May-2017.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝐷 → ran 𝐹 ⊆ 𝐷)
Theorem	mptssALT 32931*	Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 6035. (Contributed by Thierry Arnoux, 30-May-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) ⊆ (𝑥 ∈ 𝐵 ↦ 𝐶))
Theorem	dfcnv2 32932*	Alternative definition of the converse of a relation. (Contributed by Thierry Arnoux, 31-Mar-2018.)
⊢ (ran 𝑅 ⊆ 𝐴 → ◡𝑅 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})))
Theorem	partfun2 32933*	Rewrite a function defined by parts, using a mapping and an if construct, into a union of functions on disjoint domains. See also partfun 6672 and ifmpt2v 7502. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ 𝜑}    ⇒   ⊢ (𝑥 ∈ 𝐴 ↦ if(𝜑, 𝐵, 𝐶)) = ((𝑥 ∈ 𝐷 ↦ 𝐵) ∪ (𝑥 ∈ (𝐴 ∖ 𝐷) ↦ 𝐶))
Theorem	rnressnsn 32934	The range of a restriction to a singleton is a singleton. See dmressnsn 6013. (Contributed by Thierry Arnoux, 25-Jan-2026.)
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ran (𝐹 ↾ {𝐴}) = {(𝐹‘𝐴)})
21.3.4.3  Operations - misc additions
Theorem	mpomptxf 32935*	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.)
⊢ Ⅎ𝑥𝐶    &   ⊢ Ⅎ𝑦𝐶    &   ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)    ⇒   ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)
Theorem	of0r 32936	Function operation with the empty function. (Contributed by Thierry Arnoux, 27-May-2025.)
⊢ (𝐹 ∘f 𝑅∅) = ∅
21.3.4.4  The mapping operation
Theorem	elmaprd 32937	Deduction associated with elmapd 8825. Reverse direction of elmapdd 8826. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m 𝐴))    ⇒   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
21.3.4.5  Support of a function
Theorem	suppovss 32938*	A bound for the support of an operation. (Contributed by Thierry Arnoux, 19-Jul-2023.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑦 ∈ 𝐵 ↦ 𝐶))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐷)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐷)    ⇒   ⊢ (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐺 supp (𝐵 × {𝑍})) × ∪ 𝑘 ∈ (𝐺 supp (𝐵 × {𝑍}))((𝐺‘𝑘) supp 𝑍)))
Theorem	elsuppfnd 32939	Deduce membership in the support of a function. (Contributed by Thierry Arnoux, 5-Oct-2025.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐹‘𝑋) ≠ 𝑍)    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐹 supp 𝑍))
Theorem	fisuppov1 32940*	Formula building theorem for finite support: operator with left annihilator. (Contributed by Thierry Arnoux, 5-Oct-2025.)
⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → 0 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐷 ⊆ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐸)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → ( 0 𝑂𝑦) = 𝑍)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐷 ↦ ((𝐹‘𝑥)𝑂𝐵)) finSupp 𝑍)
Theorem	suppun2 32941	The support of a union is the union of the supports. (Contributed by Thierry Arnoux, 5-Oct-2025.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ((𝐹 ∪ 𝐺) supp 𝑍) = ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))
Theorem	fdifsupp 32942	Express the support of a function 𝐹 outside of 𝐵 in two different ways. (Contributed by Thierry Arnoux, 5-Oct-2025.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    ⇒   ⊢ (𝜑 → ((𝐹 ↾ (𝐴 ∖ 𝐵)) supp 𝑍) = ((𝐹 supp 𝑍) ∖ 𝐵))
Theorem	suppiniseg 32943	Relation between the support (𝐹 supp 𝑍) and the initial segment (◡𝐹 “ {𝑍}). (Contributed by Thierry Arnoux, 25-Jun-2024.)
⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (dom 𝐹 ∖ (𝐹 supp 𝑍)) = (◡𝐹 “ {𝑍}))
Theorem	fsuppinisegfi 32944	The initial segment (◡𝐹 “ {𝑌}) of a nonzero 𝑌 is finite if 𝐹 has finite support. (Contributed by Thierry Arnoux, 21-Jun-2024.)
⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → 0 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑌 ∈ (V ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (◡𝐹 “ {𝑌}) ∈ Fin)
Theorem	fressupp 32945	The restriction of a function to its support. (Contributed by Thierry Arnoux, 25-Jun-2024.)
⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 ↾ (𝐹 supp 𝑍)) = (𝐹 ∖ (V × {𝑍})))
Theorem	fdifsuppconst 32946	A function is a zero constant outside of its support. (Contributed by Thierry Arnoux, 22-Jun-2024.)
⊢ 𝐴 = (dom 𝐹 ∖ (𝐹 supp 𝑍))    ⇒   ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 ↾ 𝐴) = (𝐴 × {𝑍}))
Theorem	ressupprn 32947	The range of a function restricted to its support. (Contributed by Thierry Arnoux, 25-Jun-2024.)
⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → ran (𝐹 ↾ (𝐹 supp 0 )) = (ran 𝐹 ∖ { 0 }))
Theorem	supppreima 32948	Express the support of a function as the preimage of its range except zero. (Contributed by Thierry Arnoux, 24-Jun-2024.)
⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝐹 supp 𝑍) = (◡𝐹 “ (ran 𝐹 ∖ {𝑍})))
Theorem	fsupprnfi 32949	Finite support implies finite range. (Contributed by Thierry Arnoux, 24-Jun-2024.)
⊢ (((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) ∧ ( 0 ∈ 𝑊 ∧ 𝐹 finSupp 0 )) → ran 𝐹 ∈ Fin)
Theorem	mptiffisupp 32950*	Conditions for a mapping function defined with a conditional to have finite support. (Contributed by Thierry Arnoux, 20-Feb-2025.)
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ if(𝑥 ∈ 𝐵, 𝐶, 𝑍))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐹 finSupp 𝑍)
21.3.4.6  Explicit Functions with one or two points as a domain
Theorem	cosnopne 32951	Composition of two ordered pair singletons with non-matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐷)    ⇒   ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐷⟩}) = ∅)
Theorem	cosnop 32952	Composition of two ordered pair singletons with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑋)    ⇒   ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∘ {⟨𝐶, 𝐴⟩}) = {⟨𝐶, 𝐵⟩})
Theorem	cnvprop 32953	Converse of a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {⟨𝐵, 𝐴⟩, ⟨𝐷, 𝐶⟩})
Theorem	brprop 32954	Binary relation for a pair of ordered pairs. (Contributed by Thierry Arnoux, 24-Sep-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝑋{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}𝑌 ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷))))
Theorem	mptprop 32955*	Rewrite pairs of ordered pairs as mapping to functions. (Contributed by Thierry Arnoux, 24-Sep-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    ⇒   ⊢ (𝜑 → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
Theorem	coprprop 32956	Composition of two pairs of ordered pairs with matching domain and range. (Contributed by Thierry Arnoux, 24-Sep-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐶 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐸 ≠ 𝐹)    ⇒   ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∘ {⟨𝐸, 𝐴⟩, ⟨𝐹, 𝐶⟩}) = {⟨𝐸, 𝐵⟩, ⟨𝐹, 𝐷⟩})
Theorem	fmptunsnop 32957*	Two ways to express a function with a value replaced. (Contributed by Thierry Arnoux, 5-Oct-2025.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}))
21.3.4.7  Isomorphisms - misc. additions
Theorem	gtiso 32958	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)))
Theorem	isoun 32959*	Infer an isomorphism from a union of two isomorphisms. (Contributed by Thierry Arnoux, 30-Mar-2017.)
⊢ (𝜑 → 𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   ⊢ (𝜑 → 𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) → 𝑥𝑅𝑦)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) → 𝑧𝑆𝑤)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐴) → ¬ 𝑥𝑅𝑦)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷 ∧ 𝑤 ∈ 𝐵) → ¬ 𝑧𝑆𝑤)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅)    &   ⊢ (𝜑 → (𝐵 ∩ 𝐷) = ∅)    ⇒   ⊢ (𝜑 → (𝐻 ∪ 𝐺) Isom 𝑅, 𝑆 ((𝐴 ∪ 𝐶), (𝐵 ∪ 𝐷)))
21.3.4.8  Disjointness (additional proof requiring functions)
Theorem	disjdsct 32960*	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 6594) (Contributed by Thierry Arnoux, 28-Feb-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ (𝑉 ∖ {∅}))    &   ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)    ⇒   ⊢ (𝜑 → Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵))
21.3.4.9  First and second members of an ordered pair - misc additions
Theorem	df1stres 32961*	Definition for a restriction of the 1st (first member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
⊢ (1st ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑥)
Theorem	df2ndres 32962*	Definition for a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
⊢ (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝑦)
Theorem	1stpreimas 32963	The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.)
⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → (◡(1st ↾ 𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋})))
Theorem	1stpreima 32964	The preimage by 1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
⊢ (𝐴 ⊆ 𝐵 → (◡(1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))
Theorem	2ndpreima 32965	The preimage by 2nd is an 'horizontal band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
⊢ (𝐴 ⊆ 𝐶 → (◡(2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))
Theorem	curry2ima 32966*	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) ∧ 𝐶 ∈ 𝐵 ∧ 𝐷 ⊆ 𝐴) → (𝐺 “ 𝐷) = {𝑦 ∣ ∃𝑥 ∈ 𝐷 𝑦 = (𝑥𝐹𝐶)})
Theorem	preiman0 32967	The preimage of a nonempty set is nonempty. (Contributed by Thierry Arnoux, 9-Jun-2024.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ 𝐴) ≠ ∅)
Theorem	intimafv 32968*	The intersection of an image set, as an indexed intersection of function values. (Contributed by Thierry Arnoux, 15-Jun-2024.)
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ∩ (𝐹 “ 𝐴) = ∩ 𝑥 ∈ 𝐴 (𝐹‘𝑥))
21.3.4.10  Countable Sets
Theorem	snct 32969	A singleton is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ≼ ω)
Theorem	prct 32970	An unordered pair is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ≼ ω)
Theorem	mpocti 32971*	An operation is countable if both its domains are countable. (Contributed by Thierry Arnoux, 17-Sep-2017.)
⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑉    ⇒   ⊢ ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ≼ ω)
Theorem	abrexct 32972*	An image set of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
⊢ (𝐴 ≼ ω → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≼ ω)
Theorem	mptctf 32973	A countable mapping set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ≼ ω → (𝑥 ∈ 𝐴 ↦ 𝐵) ≼ ω)
Theorem	abrexctf 32974*	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.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ≼ ω → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≼ ω)
Theorem	padct 32975*	Index a countable set with integers and pad with 𝑍. (Contributed by Thierry Arnoux, 1-Jun-2020.) Avoid ax-rep 5232. (Revised by GG, 2-Apr-2026.)
⊢ ((𝐴 ≼ ω ∧ 𝑍 ∈ 𝑉 ∧ ¬ 𝑍 ∈ 𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (◡𝑓 ↾ 𝐴)))
Theorem	f1od2 32976*	Sufficient condition for a binary function expressed in maps-to notation to be bijective. (Contributed by Thierry Arnoux, 17-Aug-2017.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐷) → (𝐼 ∈ 𝑋 ∧ 𝐽 ∈ 𝑌))    &   ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧 ∈ 𝐷 ∧ (𝑥 = 𝐼 ∧ 𝑦 = 𝐽))))    ⇒   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)–1-1-onto→𝐷)
Theorem	fcobij 32977*	Composing functions with a bijection yields a bijection between sets of functions. (Contributed by Thierry Arnoux, 25-Aug-2017.)
⊢ (𝜑 → 𝐺:𝑆–1-1-onto→𝑇)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (𝑓 ∈ (𝑆 ↑m 𝑅) ↦ (𝐺 ∘ 𝑓)):(𝑆 ↑m 𝑅)–1-1-onto→(𝑇 ↑m 𝑅))
Theorem	fcobijfs 32978*	Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfien 9356. (Contributed by Thierry Arnoux, 25-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
⊢ (𝜑 → 𝐺:𝑆–1-1-onto→𝑇)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑆)    &   ⊢ 𝑄 = (𝐺‘𝑂)    &   ⊢ 𝑋 = {𝑔 ∈ (𝑆 ↑m 𝑅) ∣ 𝑔 finSupp 𝑂}    &   ⊢ 𝑌 = {ℎ ∈ (𝑇 ↑m 𝑅) ∣ ℎ finSupp 𝑄}    ⇒   ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌)
Theorem	fcobijfs2 32979*	Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also fcobijfs 32978 and mapfien 9356. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ (𝜑 → 𝐺:𝑅–1-1-onto→𝑆)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑂 ∈ 𝑇)    &   ⊢ 𝑋 = {𝑔 ∈ (𝑇 ↑m 𝑆) ∣ 𝑔 finSupp 𝑂}    &   ⊢ 𝑌 = {ℎ ∈ (𝑇 ↑m 𝑅) ∣ ℎ finSupp 𝑂}    ⇒   ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝑓 ∘ 𝐺)):𝑋–1-1-onto→𝑌)
Theorem	suppss3 32980*	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 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = 𝑍) → 𝐵 = 𝑍)    ⇒   ⊢ (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))
Theorem	fsuppcurry1 32981*	Finite support of a curried function with a constant first argument. (Contributed by Thierry Arnoux, 7-Jul-2023.)
⊢ 𝐺 = (𝑥 ∈ 𝐵 ↦ (𝐶𝐹𝑥))    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 Fn (𝐴 × 𝐵))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐹 finSupp 𝑍)    ⇒   ⊢ (𝜑 → 𝐺 finSupp 𝑍)
Theorem	fsuppcurry2 32982*	Finite support of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 7-Jul-2023.)
⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝑥𝐹𝐶))    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 Fn (𝐴 × 𝐵))    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 𝑍)    ⇒   ⊢ (𝜑 → 𝐺 finSupp 𝑍)
Theorem	offinsupp1 32983*	Finite support for a function operation. (Contributed by Thierry Arnoux, 8-Jul-2023.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝑇)    &   ⊢ (𝜑 → 𝐹 finSupp 𝑌)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑇) → (𝑌𝑅𝑥) = 𝑍)    ⇒   ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) finSupp 𝑍)
Theorem	ffs2 32984	Rewrite a function's support based with its codomain rather than the universal class. See also fsuppeq 8159. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
⊢ 𝐶 = (𝐵 ∖ {𝑍})    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 supp 𝑍) = (◡𝐹 “ 𝐶))
Theorem	ffsrn 32985	The range of a finitely supported function is finite. (Contributed by Thierry Arnoux, 27-Aug-2017.)
⊢ (𝜑 → 𝑍 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → (𝐹 supp 𝑍) ∈ Fin)    ⇒   ⊢ (𝜑 → ran 𝐹 ∈ Fin)
Theorem	cocnvf1o 32986	Composing with the inverse of a bijection. (Contributed by Thierry Arnoux, 15-Jan-2026.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐻:𝐴–1-1-onto→𝐴)    ⇒   ⊢ (𝜑 → (𝐹 = (𝐺 ∘ 𝐻) ↔ 𝐺 = (𝐹 ∘ ◡𝐻)))
Theorem	resf1o 32987*	Restriction of functions to a superset of their support creates a bijection. (Contributed by Thierry Arnoux, 12-Sep-2017.)
⊢ 𝑋 = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ (◡𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶}    &   ⊢ 𝐹 = (𝑓 ∈ 𝑋 ↦ (𝑓 ↾ 𝐶))    ⇒   ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ⊆ 𝐴) ∧ 𝑍 ∈ 𝐵) → 𝐹:𝑋–1-1-onto→(𝐵 ↑m 𝐶))
Theorem	maprnin 32988*	Restricting the range of the mapping operator. (Contributed by Thierry Arnoux, 30-Aug-2017.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝐵 ∩ 𝐶) ↑m 𝐴) = {𝑓 ∈ (𝐵 ↑m 𝐴) ∣ ran 𝑓 ⊆ 𝐶}
Theorem	fpwrelmapffslem 32989*	Lemma for fpwrelmapffs 32991. 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    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝒫 𝐵)    &   ⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝐹‘𝑥))})    ⇒   ⊢ (𝜑 → (𝑅 ∈ Fin ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))
Theorem	fpwrelmap 32990*	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 10420 and marypha2lem1 9383. (Contributed by Thierry Arnoux, 28-Aug-2017.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝑀 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})    ⇒   ⊢ 𝑀:(𝒫 𝐵 ↑m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)
Theorem	fpwrelmapffs 32991*	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 𝐵 ↑m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})    &   ⊢ 𝑆 = {𝑓 ∈ ((𝒫 𝐵 ∩ Fin) ↑m 𝐴) ∣ (𝑓 supp ∅) ∈ Fin}    ⇒   ⊢ (𝑀 ↾ 𝑆):𝑆–1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin)
21.3.5  Real and Complex Numbers
Theorem	sgnval2 32992	Value of the signum of a real number, expresssed using absolute value. (Contributed by Thierry Arnoux, 9-Nov-2025.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0) → (sgn‘𝐴) = (𝐴 / (abs‘𝐴)))
21.3.5.1  Complex operations - misc. additions
Theorem	creq0 32993	The real representation of complex numbers is zero iff both its terms are zero. Cf. crne0 12202. (Contributed by Thierry Arnoux, 20-Aug-2023.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 = 0 ∧ 𝐵 = 0) ↔ (𝐴 + (i · 𝐵)) = 0))
Theorem	1nei 32994	The imaginary unit i is not one. (Contributed by Thierry Arnoux, 20-Aug-2023.)
⊢ 1 ≠ i
Theorem	1neg1t1neg1 32995	An integer unit times itself. (Contributed by Thierry Arnoux, 23-Aug-2020.)
⊢ (𝑁 ∈ {-1, 1} → (𝑁 · 𝑁) = 1)
Theorem	nnmulge 32996	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))
Theorem	submuladdd 32997	The product of a difference and a sum. Cf. addmulsub 11664. (Contributed by Thierry Arnoux, 6-Jul-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) · (𝐶 + 𝐷)) = (((𝐴 · 𝐶) + (𝐴 · 𝐷)) − ((𝐵 · 𝐶) + (𝐵 · 𝐷))))
Theorem	binom2subadd 32998	The difference of the squares of the sum and difference of two complex numbers 𝐴 and 𝐵. (Contributed by Thierry Arnoux, 5-Nov-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (((𝐴 + 𝐵)↑2) − ((𝐴 − 𝐵)↑2)) = (4 · (𝐴 · 𝐵)))
Theorem	cjsubd 32999	Complex conjugate distributes over subtraction. (Contributed by Thierry Arnoux, 1-Jul-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → (∗‘(𝐴 − 𝐵)) = ((∗‘𝐴) − (∗‘𝐵)))
Theorem	re0cj 33000	The conjugate of a pure imaginary number is its negative. (Contributed by Thierry Arnoux, 25-Jun-2025.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → (ℜ‘𝐴) = 0)    ⇒   ⊢ (𝜑 → (∗‘𝐴) = -𝐴)