P. 327 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 32601-32700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	iunsnima 32601	Image of a singleton by an indexed union involving that singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑥}) = 𝐵)
Theorem	iunsnima2 32602*	Version of iunsnima 32601 with different variables. (Contributed by Thierry Arnoux, 22-Jun-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑊)    &   ⊢ Ⅎ𝑥𝐶    &   ⊢ (𝑥 = 𝑌 → 𝐵 = 𝐶)    ⇒   ⊢ ((𝜑 ∧ 𝑌 ∈ 𝐴) → (∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) “ {𝑌}) = 𝐶)
21.3.4.2  Functions - misc additions
Theorem	fconst7v 32603*	An alternative way to express a constant function. (Contributed by Glauco Siliprandi, 5-Feb-2022.) Removed hyphotheses as suggested by SN (Revised by Thierry Arnoux, 10-Jan-2026.)
⊢ (𝜑 → 𝐹 Fn 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)    ⇒   ⊢ (𝜑 → 𝐹 = (𝐴 × {𝐵}))
Theorem	constcof 32604	Composition with a constant function. See also fcoconst 7067. (Contributed by Thierry Arnoux, 11-Jan-2026.)
⊢ (𝜑 → 𝐹:𝑋⟶𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐼 × {𝑌}) ∘ 𝐹) = (𝑋 × {𝑌}))
Theorem	ac6sf2 32605*	Alternate version of ac6 10371 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.) (Revised by Thierry Arnoux, 17-May-2020.)
⊢ Ⅎ𝑦𝐵    &   ⊢ Ⅎ𝑦𝜓    &   ⊢ 𝐴 ∈ V    &   ⊢ (𝑦 = (𝑓‘𝑥) → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑓(𝑓:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 𝜓))
Theorem	ac6mapd 32606*	Axiom of choice equivalent, deduction form. (Contributed by Thierry Arnoux, 13-Oct-2025.)
⊢ (𝑦 = (𝑓‘𝑥) → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝜓)    ⇒   ⊢ (𝜑 → ∃𝑓 ∈ (𝐵 ↑m 𝐴)∀𝑥 ∈ 𝐴 𝜒)
Theorem	fnresin 32607	Restriction of a function with a subclass of its domain. (Contributed by Thierry Arnoux, 10-Oct-2017.)
⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐵) Fn (𝐴 ∩ 𝐵))
Theorem	f1o3d 32608*	Describe an implicit one-to-one onto function. (Contributed by Thierry Arnoux, 23-Apr-2017.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐷 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 = 𝐷 ↔ 𝑦 = 𝐶))    ⇒   ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ 𝐷)))
Theorem	eldmne0 32609	A function of nonempty domain is not empty. (Contributed by Thierry Arnoux, 20-Nov-2023.)
⊢ (𝑋 ∈ dom 𝐹 → 𝐹 ≠ ∅)
Theorem	f1rnen 32610	Equinumerosity of the range of an injective function. (Contributed by Thierry Arnoux, 7-Jul-2023.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉) → ran 𝐹 ≈ 𝐴)
Theorem	f1oeq3dd 32611	Equality deduction for one-to-one onto functions. (Contributed by Thierry Arnoux, 10-Jan-2026.)
⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐵)
Theorem	rinvf1o 32612	Sufficient conditions for the restriction of an involution to be a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
⊢ Fun 𝐹    &   ⊢ ◡𝐹 = 𝐹    &   ⊢ (𝐹 “ 𝐴) ⊆ 𝐵    &   ⊢ (𝐹 “ 𝐵) ⊆ 𝐴    &   ⊢ 𝐴 ⊆ dom 𝐹    &   ⊢ 𝐵 ⊆ dom 𝐹    ⇒   ⊢ (𝐹 ↾ 𝐴):𝐴–1-1-onto→𝐵
Theorem	fresf1o 32613	Conditions for a restriction to be a one-to-one onto function. (Contributed by Thierry Arnoux, 7-Dec-2016.)
⊢ ((Fun 𝐹 ∧ 𝐶 ⊆ ran 𝐹 ∧ Fun (◡𝐹 ↾ 𝐶)) → (𝐹 ↾ (◡𝐹 “ 𝐶)):(◡𝐹 “ 𝐶)–1-1-onto→𝐶)
Theorem	nfpconfp 32614	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 ))
Theorem	fmptco1f1o 32615*	The action of composing (to the right) with a bijection is itself a bijection of functions. (Contributed by Thierry Arnoux, 3-Jan-2021.)
⊢ 𝐴 = (𝑅 ↑m 𝐸)    &   ⊢ 𝐵 = (𝑅 ↑m 𝐷)    &   ⊢ 𝐹 = (𝑓 ∈ 𝐴 ↦ (𝑓 ∘ 𝑇))    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐸 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑇:𝐷–1-1-onto→𝐸)    ⇒   ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)
Theorem	cofmpt2 32616*	Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 15-Jul-2023.)
⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐶 = 𝐷)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → 𝐶 ∈ 𝐸)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑦 ∈ 𝐵 ↦ 𝐶) ∘ 𝐹) = (𝑥 ∈ 𝐴 ↦ 𝐷))
Theorem	f1mptrn 32617*	Express injection for a mapping operation. (Contributed by Thierry Arnoux, 3-May-2020.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 𝑦 = 𝐵)    ⇒   ⊢ (𝜑 → Fun ◡(𝑥 ∈ 𝐴 ↦ 𝐵))
Theorem	dfimafnf 32618*	Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐹    ⇒   ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
Theorem	funimass4f 32619	Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐹    ⇒   ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
Theorem	suppss2f 32620*	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.)
⊢ Ⅎ𝑘𝜑    &   ⊢ Ⅎ𝑘𝐴    &   ⊢ Ⅎ𝑘𝑊    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝑊)) → 𝐵 = 𝑍)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) supp 𝑍) ⊆ 𝑊)
Theorem	ofrn 32621	The range of the function operation. (Contributed by Thierry Arnoux, 8-Jan-2017.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ 𝐶)
Theorem	ofrn2 32622	The range of the function operation. (Contributed by Thierry Arnoux, 21-Mar-2017.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐶)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ran (𝐹 ∘f + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))
Theorem	off2 32623*	The function operation produces a function - alternative form with all antecedents as deduction. (Contributed by Thierry Arnoux, 17-Feb-2017.)
⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)    &   ⊢ (𝜑 → 𝐺:𝐵⟶𝑇)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈)
Theorem	ofresid 32624	Applying an operation restricted to the range of the functions does not change the function operation. (Contributed by Thierry Arnoux, 14-Feb-2018.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐺:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝐹 ∘f (𝑅 ↾ (𝐵 × 𝐵))𝐺))
Theorem	unipreima 32625*	Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.)
⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝐴) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥))
Theorem	opfv 32626	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 32627	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 32628	Image of a cartesian product by 2nd. (Contributed by Thierry Arnoux, 23-Jun-2024.)
⊢ (𝐴 ≠ ∅ → (2nd “ (𝐴 × 𝐵)) = 𝐵)
Theorem	dmdju 32629*	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 32630*	Stronger version of djussxp 5784. (Contributed by Thierry Arnoux, 23-Jun-2024.)
⊢ ∪ 𝑘 ∈ 𝐴 ({𝑘} × 𝐵) ⊆ (𝐴 × ∪ 𝑘 ∈ 𝐴 𝐵)
Theorem	2ndresdju 32631*	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 32632*	The 2nd function restricted to a disjoint union is a bijection. See also e.g. 2ndconst 8031. (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 32633*	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 32634*	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 32635*	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 32636*	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 32637*	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 32638	Domain and codomain of the mapping operation; deduction form. This version of fmptd 7047 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 32639*	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 32640*	Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 7066. (Contributed by Thierry Arnoux, 30-Jun-2017.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ ((𝐴:𝐷⟶𝐸 ∧ 𝐵:𝐶⟶𝐷) → (𝐴 ∘ 𝐵) = (𝑥 ∈ 𝐶 ↦ (𝐴‘(𝐵‘𝑥))))
Theorem	acunirnmpt 32641*	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 32642*	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 32643*	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 32644*	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 32645*	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 32646*	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 32647*	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 32648*	Express the preimage of a function operation as a union of preimages. This version of ofpreima 32647 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	funcnvmpt 32649*	Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (Fun ◡𝐹 ↔ ∀𝑦∃*𝑥 ∈ 𝐴 𝑦 = 𝐵))
Theorem	funcnv5mpt 32650*	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 32651*	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 32652	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 32653*	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 32654*	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 32655*	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 32656*	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 32657*	Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 5990. (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 32658*	Alternative definition of the converse of a relation. (Contributed by Thierry Arnoux, 31-Mar-2018.)
⊢ (ran 𝑅 ⊆ 𝐴 → ◡𝑅 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})))
21.3.4.3  Operations - misc additions
Theorem	mpomptxf 32659*	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 32660	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 32661	Deduction associated with elmapd 8764. Reverse direction of elmapdd 8765. (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 32662*	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 32663	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 32664*	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 32665	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 32666	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 32667	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 32668	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 32669	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 32670	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 32671	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 32672	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 32673	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 32674*	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 32675	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 32676	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 32677	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 32678	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 32679*	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 32680	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 32681*	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 32682	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 32683*	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 32684*	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 6550) (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 32685*	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 32686*	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 32687	The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.)
⊢ ((Rel 𝐴 ∧ 𝑋 ∈ 𝑉) → (◡(1st ↾ 𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋})))
Theorem	1stpreima 32688	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 32689	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 32690*	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 32691	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 32692*	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  Finite Sets
Theorem	imafi2 32693	The image by a finite set is finite. See also imafi 9199. (Contributed by Thierry Arnoux, 25-Apr-2020.)
⊢ (𝐴 ∈ Fin → (𝐴 “ 𝐵) ∈ Fin)
Theorem	unifi3 32694	If a union is finite, then all its elements are finite. See unifi 9228. (Contributed by Thierry Arnoux, 27-Aug-2017.)
⊢ (∪ 𝐴 ∈ Fin → 𝐴 ⊆ Fin)
21.3.4.11  Countable Sets
Theorem	snct 32695	A singleton is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.)
⊢ (𝐴 ∈ 𝑉 → {𝐴} ≼ ω)
Theorem	prct 32696	An unordered pair is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {𝐴, 𝐵} ≼ ω)
Theorem	mpocti 32697*	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 32698*	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 32699	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 32700*	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)