P. 494 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 49301-49400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	disjdifb 49301	Relative complement is anticommutative regarding intersection. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅
Theorem	predisj 49302	Preimages of disjoint sets are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (𝜑 → Fun 𝐹)    &   ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)    &   ⊢ (𝜑 → 𝑆 ⊆ (◡𝐹 “ 𝐴))    &   ⊢ (𝜑 → 𝑇 ⊆ (◡𝐹 “ 𝐵))    ⇒   ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅)
21.49.3.4  Unordered and ordered pairs
Theorem	vsn 49303	The singleton of the universal class is the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ {V} = ∅
Theorem	mosn 49304*	"At most one" element in a singleton. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
Theorem	mo0 49305*	"At most one" element in an empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)
Theorem	mosssn 49306*	"At most one" element in a subclass of a singleton. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
Theorem	mo0sn 49307*	Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ (𝐴 = ∅ ∨ ∃𝑦 𝐴 = {𝑦}))
Theorem	mosssn2 49308*	Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦})
21.49.3.5  The union of a class
Theorem	unilbss 49309*	Superclass of the greatest lower bound. A dual statement of ssintub 4909. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴
21.49.3.6  Indexed union and intersection
Theorem	iuneq0 49310	An indexed union is empty iff all indexed classes are empty. (Contributed by Zhi Wang, 1-Nov-2025.)
⊢ (∀𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ∪ 𝑥 ∈ 𝐴 𝐵 = ∅)
Theorem	iineq0 49311	An indexed intersection is empty if one of the intersected classes is empty. (Contributed by Zhi Wang, 30-Oct-2025.)
⊢ (∃𝑥 ∈ 𝐴 𝐵 = ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = ∅)
Theorem	iunlub 49312*	The indexed union is the the lowest upper bound if it exists. (Contributed by Zhi Wang, 1-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶)    ⇒   ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)
Theorem	iinglb 49313*	The indexed intersection is the the greatest lower bound if it exists. (Contributed by Zhi Wang, 1-Nov-2025.)
⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → 𝐵 = 𝐶)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ⊆ 𝐵)    ⇒   ⊢ (𝜑 → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐶)
Theorem	iuneqconst2 49314*	Indexed union of identical classes. (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)
Theorem	iineqconst2 49315*	Indexed intersection of identical classes. (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐶)
21.49.4  ZF Set Theory - add the Axiom of Replacement
21.49.4.1  Theorems requiring subset and intersection existence
Theorem	inpw 49316*	Two ways of expressing a collection of subsets as seen in df-ntr 22999, unimax 4888, and others. (Contributed by Zhi Wang, 27-Sep-2024.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵})
21.49.5  ZF Set Theory - add the Axiom of Power Sets
21.49.5.1  Ordered pair theorem
Theorem	opth1neg 49317	Two ordered pairs are not equal if their first components are not equal. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ≠ 𝐶 → ⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩))
Theorem	opth2neg 49318	Two ordered pairs are not equal if their second components are not equal. (Contributed by Zhi Wang, 7-Oct-2025.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ≠ 𝐷 → ⟨𝐴, 𝐵⟩ ≠ ⟨𝐶, 𝐷⟩))
21.49.5.2  Ordered-pair class abstractions (cont.)
Theorem	brab2dd 49319*	Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by Zhi Wang, 24-Sep-2025.)
⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓)})    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ↔ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)))    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))
Theorem	brab2ddw 49320*	Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by Zhi Wang, 24-Sep-2025.)
⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓)})    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑦 = 𝐵 → (𝜃 ↔ 𝜒))    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐶 = 𝑈)    &   ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝐷 = 𝑉)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))
Theorem	brab2ddw2 49321*	Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by Zhi Wang, 24-Sep-2025.)
⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜓)})    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑦 = 𝐵 → (𝜃 ↔ 𝜒))    &   ⊢ (𝑥 = 𝐴 → 𝐶 = 𝑈)    &   ⊢ (𝑦 = 𝐵 → 𝐷 = 𝑉)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉) ∧ 𝜒)))
21.49.5.3  Relations
Theorem	iinxp 49322*	Indexed intersection of Cartesian products is the Cartesian product of indexed intersections. See also inxp 5782 and intxp 49323. (Contributed by Zhi Wang, 30-Oct-2025.)
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) = (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶))
Theorem	intxp 49323*	Intersection of Cartesian products is the Cartesian product of intersection of domains and ranges. See also inxp 5782 and iinxp 49322. (Contributed by Zhi Wang, 30-Oct-2025.)
⊢ (𝜑 → 𝐴 ≠ ∅)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = (dom 𝑥 × ran 𝑥))    &   ⊢ 𝑋 = ∩ 𝑥 ∈ 𝐴 dom 𝑥    &   ⊢ 𝑌 = ∩ 𝑥 ∈ 𝐴 ran 𝑥    ⇒   ⊢ (𝜑 → ∩ 𝐴 = (𝑋 × 𝑌))
Theorem	coxp 49324	Composition with a Cartesian product. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (𝐴 ∘ (𝐵 × 𝐶)) = (𝐵 × (𝐴 “ 𝐶))
Theorem	cosn 49325	Composition with an ordered pair singleton. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ ((𝐵 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉) → (𝐴 ∘ {⟨𝐵, 𝐶⟩}) = ({𝐵} × (𝐴 “ {𝐶})))
Theorem	cosni 49326	Composition with an ordered pair singleton. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴 ∘ {⟨𝐵, 𝐶⟩}) = ({𝐵} × (𝐴 “ {𝐶}))
Theorem	inisegn0a 49327	The inverse image of a singleton subset of an image is non-empty. (Contributed by Zhi Wang, 7-Nov-2025.)
⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (◡𝐹 “ {𝐴}) ≠ ∅)
Theorem	dmrnxp 49328	A Cartesian product is the Cartesian product of its domain and range. (Contributed by Zhi Wang, 30-Oct-2025.)
⊢ (𝑅 = (𝐴 × 𝐵) → 𝑅 = (dom 𝑅 × ran 𝑅))
21.49.5.4  Functions
Theorem	mof0 49329	There is at most one function into the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ ∃*𝑓 𝑓:𝐴⟶∅
Theorem	mof02 49330*	A variant of mof0 49329. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵)
Theorem	mof0ALT 49331*	Alternate proof of mof0 49329 with stronger requirements on distinct variables. Uses mo4 2567. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃*𝑓 𝑓:𝐴⟶∅
Theorem	eufsnlem 49332*	There is exactly one function into a singleton. For a simpler hypothesis, see eufsn 49333 assuming ax-rep 5213, or eufsn2 49334 assuming ax-pow 5304 and ax-un 7684. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → (𝐴 × {𝐵}) ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Theorem	eufsn 49333*	There is exactly one function into a singleton, assuming ax-rep 5213. See eufsn2 49334 for different axiom requirements. If existence is not needed, use mofsn 49335 or mofsn2 49336 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Theorem	eufsn2 49334*	There is exactly one function into a singleton, assuming ax-pow 5304 and ax-un 7684. Variant of eufsn 49333. If existence is not needed, use mofsn 49335 or mofsn2 49336 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Theorem	mofsn 49335*	There is at most one function into a singleton, with fewer axioms than eufsn 49333 and eufsn2 49334. See also mofsn2 49336. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐵 ∈ 𝑉 → ∃*𝑓 𝑓:𝐴⟶{𝐵})
Theorem	mofsn2 49336*	There is at most one function into a singleton. An unconditional variant of mofsn 49335, i.e., the singleton could be empty if 𝑌 is a proper class. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐵 = {𝑌} → ∃*𝑓 𝑓:𝐴⟶𝐵)
Theorem	mofsssn 49337*	There is at most one function into a subclass of a singleton. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝐵 ⊆ {𝑌} → ∃*𝑓 𝑓:𝐴⟶𝐵)
Theorem	mofmo 49338*	There is at most one function into a class containing at most one element. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (∃*𝑥 𝑥 ∈ 𝐵 → ∃*𝑓 𝑓:𝐴⟶𝐵)
Theorem	mofeu 49339*	The uniqueness of a function into a set with at most one element. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ 𝐺 = (𝐴 × 𝐵)    &   ⊢ (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))    &   ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ↔ 𝐹 = 𝐺))
Theorem	elfvne0 49340	If a function value has a member, then the function is not an empty set (An artifact of our function value definition.) (Contributed by Zhi Wang, 16-Sep-2024.)
⊢ (𝐴 ∈ (𝐹‘𝐵) → 𝐹 ≠ ∅)
Theorem	fdomne0 49341	A function with non-empty domain is non-empty and has non-empty codomain. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑋 ≠ ∅) → (𝐹 ≠ ∅ ∧ 𝑌 ≠ ∅))
Theorem	f1sn2g 49342	A function that maps a singleton to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:{𝐴}⟶𝐵) → 𝐹:{𝐴}–1-1→𝐵)
Theorem	f102g 49343	A function that maps the empty set to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ ((𝐴 = ∅ ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
Theorem	f1mo 49344*	A function that maps a set with at most one element to a class is injective. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ ((∃*𝑥 𝑥 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 𝐹:𝐴–1-1→𝐵)
Theorem	f002 49345	A function with an empty codomain must have empty domain. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    ⇒   ⊢ (𝜑 → (𝐵 = ∅ → 𝐴 = ∅))
Theorem	map0cor 49346*	A function exists iff an empty codomain is accompanied with an empty domain. (Contributed by Zhi Wang, 1-Oct-2024.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → ((𝐵 = ∅ → 𝐴 = ∅) ↔ ∃𝑓 𝑓:𝐴⟶𝐵))
Theorem	ffvbr 49347	Relation with function value. (Contributed by Zhi Wang, 25-Nov-2025.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑋𝐹(𝐹‘𝑋))
Theorem	xpco2 49348	Composition of a Cartesian product with a function. (Contributed by Zhi Wang, 25-Nov-2025.)
⊢ (𝐹:𝐴⟶𝐵 → ((𝐵 × 𝐶) ∘ 𝐹) = (𝐴 × 𝐶))
21.49.5.5  Operations
Theorem	ovsng 49349	The operation value of a singleton of a nested ordered pair is the last member. (Contributed by Zhi Wang, 22-Oct-2025.)
⊢ (𝐶 ∈ 𝑉 → (𝐴{⟨⟨𝐴, 𝐵⟩, 𝐶⟩}𝐵) = 𝐶)
Theorem	ovsng2 49350	The operation value of a singleton of an ordered triple is the last member. (Contributed by Zhi Wang, 22-Oct-2025.)
⊢ (𝐶 ∈ 𝑉 → (𝐴{⟨𝐴, 𝐵, 𝐶⟩}𝐵) = 𝐶)
Theorem	ovsn 49351	The operation value of a singleton of a nested ordered pair is the last member. (Contributed by Zhi Wang, 22-Oct-2025.)
⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴{⟨⟨𝐴, 𝐵⟩, 𝐶⟩}𝐵) = 𝐶
Theorem	ovsn2 49352	The operation value of a singleton of an ordered triple is the last member. (Contributed by Zhi Wang, 22-Oct-2025.)
⊢ 𝐶 ∈ V    ⇒   ⊢ (𝐴{⟨𝐴, 𝐵, 𝐶⟩}𝐵) = 𝐶
Theorem	fvconstr 49353	Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌}))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ≠ ∅)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) = 𝑌))
Theorem	fvconstrn0 49354	Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌}))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ≠ ∅)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ (𝐴𝐹𝐵) ≠ ∅))
Theorem	fvconstr2 49355	Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐹𝐵))    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐵)
Theorem	ovmpt4d 49356*	Deduction version of ovmpt4g 7509. (This is the operation analogue of fvmpt2d 6957.) (Contributed by Zhi Wang, 9-Oct-2025.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑉)    ⇒   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐹𝑦) = 𝐶)
Theorem	eqfnovd 49357*	Deduction for equality of operations. (Contributed by Zhi Wang, 19-Nov-2025.)
⊢ (𝜑 → 𝐹 Fn (𝐴 × 𝐵))    &   ⊢ (𝜑 → 𝐺 Fn (𝐴 × 𝐵))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐹𝑦) = (𝑥𝐺𝑦))    ⇒   ⊢ (𝜑 → 𝐹 = 𝐺)
21.49.6  ZF Set Theory - add the Axiom of Union
21.49.6.1  Relations and functions (cont.)
Theorem	fonex 49358	The domain of a surjection is a proper class if the range is a proper class as well. Can be used to prove that if a structure component extractor restricted to a class maps onto a proper class, then the class is a proper class as well. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ 𝐵 ∉ V    &   ⊢ 𝐹:𝐴–onto→𝐵    ⇒   ⊢ 𝐴 ∉ V
21.49.6.2  First and second members of an ordered pair
Theorem	eloprab1st2nd 49359*	Reconstruction of a nested ordered pair in terms of its ordered pair components. (Contributed by Zhi Wang, 27-Oct-2025.)
⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝐴 = ⟨⟨(1st ‘(1st ‘𝐴)), (2nd ‘(1st ‘𝐴))⟩, (2nd ‘𝐴)⟩)
21.49.6.3  Operations in maps-to notation (continued)
Theorem	fmpodg 49360*	Domain and codomain of the mapping operation; deduction form. (Contributed by Zhi Wang, 29-Sep-2025.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑅 = (𝐴 × 𝐵))    ⇒   ⊢ (𝜑 → 𝐹:𝑅⟶𝑆)
Theorem	fmpod 49361*	Domain and codomain of the mapping operation; deduction form. (Contributed by Zhi Wang, 30-Sep-2025.)
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝑆)    ⇒   ⊢ (𝜑 → 𝐹:(𝐴 × 𝐵)⟶𝑆)
21.49.6.4  Function transposition
Theorem	resinsnlem 49362	Lemma for resinsnALT 49364. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (𝜑 → (𝜒 ↔ ¬ 𝜓))    &   ⊢ (¬ 𝜑 → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ 𝜒)
Theorem	resinsn 49363	Restriction to the intersection with a singleton. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ ((𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ (dom 𝐹 ∩ 𝐴))
Theorem	resinsnALT 49364	Restriction to the intersection with a singleton. (Contributed by Zhi Wang, 6-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ (dom 𝐹 ∩ 𝐴))
Theorem	dftpos5 49365*	Alternate definition of tpos. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ tpos 𝐹 = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))
Theorem	dftpos6 49366*	Alternate definition of tpos. The second half of the right hand side could apply ressn 6245 and become (𝐹 ↾ {∅}). (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ tpos 𝐹 = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ ({∅} × (𝐹 “ {∅})))
Theorem	dmtposss 49367	The domain of tpos 𝐹 is a subset. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})
Theorem	tposres0 49368	The transposition of a set restricted to the empty set is the set restricted to the empty set. See also ressn 6245 and dftpos6 49366 for an alternate proof. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅})
Theorem	tposresg 49369	The transposition restricted to a set. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ 𝑅) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))
Theorem	tposrescnv 49370*	The transposition restricted to a converse is the transposition of the restricted class, with the empty set removed from the domain. Note that the right hand side is a more useful form of (tpos (𝐹 ↾ 𝑅) ↾ (V ∖ {∅})) by df-tpos 8171. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ ◡𝑅) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥}))
Theorem	tposres2 49371	The transposition restricted to a set. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (𝜑 → ¬ ∅ ∈ (dom 𝐹 ∩ 𝑅))    ⇒   ⊢ (𝜑 → (tpos 𝐹 ↾ 𝑅) = (tpos 𝐹 ↾ ◡◡𝑅))
Theorem	tposres3 49372	The transposition restricted to a set. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (𝜑 → ¬ ∅ ∈ (dom 𝐹 ∩ 𝑅))    ⇒   ⊢ (𝜑 → (tpos 𝐹 ↾ 𝑅) = tpos (𝐹 ↾ ◡𝑅))
Theorem	tposres 49373	The transposition restricted to a relation. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (Rel 𝑅 → (tpos 𝐹 ↾ 𝑅) = tpos (𝐹 ↾ ◡𝑅))
Theorem	tposresxp 49374	The transposition restricted to a Cartesian product. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ (𝐴 × 𝐵)) = tpos (𝐹 ↾ (𝐵 × 𝐴))
Theorem	tposf1o 49375	Condition of a bijective transposition. (Contributed by Zhi Wang, 5-Oct-2025.)
⊢ (𝐹:(𝐴 × 𝐵)–1-1-onto→𝐶 → tpos 𝐹:(𝐵 × 𝐴)–1-1-onto→𝐶)
Theorem	tposid 49376	Swap an ordered pair. (Contributed by Zhi Wang, 5-Oct-2025.)
⊢ (𝑋tpos I 𝑌) = ⟨𝑌, 𝑋⟩
Theorem	tposidres 49377	Swap an ordered pair. (Contributed by Zhi Wang, 5-Oct-2025.)
⊢ (𝜑 → 𝑋 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑌tpos ( I ↾ (𝐴 × 𝐵))𝑋) = ⟨𝑋, 𝑌⟩)
Theorem	tposidf1o 49378	The swap function, or the twisting map, is bijective. (Contributed by Zhi Wang, 5-Oct-2025.)
⊢ tpos ( I ↾ (𝐴 × 𝐵)):(𝐵 × 𝐴)–1-1-onto→(𝐴 × 𝐵)
Theorem	tposideq 49379*	Two ways of expressing the swap function. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (Rel 𝑅 → (tpos I ↾ 𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ◡{𝑥}))
Theorem	tposideq2 49380*	Two ways of expressing the swap function. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ 𝑅 = (𝐴 × 𝐵)    ⇒   ⊢ (tpos I ↾ 𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ◡{𝑥})
21.49.6.5  Infinite Cartesian products
Theorem	ixpv 49381*	Infinite Cartesian product of the universal class is the set of functions with a fixed domain. (Contributed by Zhi Wang, 1-Nov-2025.)
⊢ X𝑥 ∈ 𝐴 V = {𝑓 ∣ 𝑓 Fn 𝐴}
21.49.6.6  Equinumerosity
Theorem	fvconst0ci 49382	A constant function's value is either the constant or the empty set. (An artifact of our function value definition.) (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ 𝐵 ∈ V    &   ⊢ 𝑌 = ((𝐴 × {𝐵})‘𝑋)    ⇒   ⊢ (𝑌 = ∅ ∨ 𝑌 = 𝐵)
Theorem	fvconstdomi 49383	A constant function's value is dominated by the constant. (An artifact of our function value definition.) (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ 𝐵 ∈ V    ⇒   ⊢ ((𝐴 × {𝐵})‘𝑋) ≼ 𝐵
Theorem	f1omo 49384*	There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Proof could be significantly shortened by fvconstdomi 49383 assuming ax-un 7684 (see f1omoALT 49386). (Contributed by Zhi Wang, 19-Sep-2024.) (Proof shortened by SN, 24-Nov-2025.)
⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))    ⇒   ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))
Theorem	f1omoOLD 49385*	Obsolete version of f1omo 49384 as of 24-Nov-2025. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))    ⇒   ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))
Theorem	f1omoALT 49386*	There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Use f1omo 49384 without assuming ax-un 7684. (Contributed by Zhi Wang, 18-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))    ⇒   ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))
21.49.7  Order sets
21.49.7.1  Real number intervals
Theorem	iccin 49387	Intersection of two closed intervals of extended reals. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = (if(𝐴 ≤ 𝐶, 𝐶, 𝐴)[,]if(𝐵 ≤ 𝐷, 𝐵, 𝐷)))
Theorem	iccdisj2 49388	If the upper bound of one closed interval is less than the lower bound of the other, the intervals are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ ((𝐴 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐵 < 𝐶) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = ∅)
Theorem	iccdisj 49389	If the upper bound of one closed interval is less than the lower bound of the other, the intervals are disjoint. (Contributed by Zhi Wang, 9-Sep-2024.)
⊢ ((((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐶 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) ∧ 𝐵 < 𝐶) → ((𝐴[,]𝐵) ∩ (𝐶[,]𝐷)) = ∅)
21.49.8  Extensible structures
21.49.8.1  Basic definitions
Theorem	slotresfo 49390*	The condition of a structure component extractor restricted to a class being a surjection. This combined with fonex 49358 can be used to prove a class being proper. (Contributed by Zhi Wang, 20-Oct-2025.)
⊢ 𝐸 Fn V    &   ⊢ (𝑘 ∈ 𝐴 → (𝐸‘𝑘) ∈ 𝑉)    &   ⊢ (𝑏 ∈ 𝑉 → 𝐾 ∈ 𝐴)    &   ⊢ (𝑏 ∈ 𝑉 → 𝑏 = (𝐸‘𝐾))    ⇒   ⊢ (𝐸 ↾ 𝐴):𝐴–onto→𝑉
21.49.9  Moore spaces
Theorem	mreuniss 49391	The union of a collection of closed sets is a subset. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → ∪ 𝑆 ⊆ 𝑋)
21.49.10  Topology Additional contents for topology.
21.49.10.1  Closure and interior
Theorem	clduni 49392	The union of closed sets is the underlying set of the topology (the union of open sets). (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝐽 ∈ Top → ∪ (Clsd‘𝐽) = ∪ 𝐽)
Theorem	opncldeqv 49393*	Conditions on open sets are equivalent to conditions on closed sets. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑥 = (∪ 𝐽 ∖ 𝑦)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐽 𝜓 ↔ ∀𝑦 ∈ (Clsd‘𝐽)𝜒))
Theorem	opndisj 49394	Two ways of saying that two open sets are disjoint, if 𝐽 is a topology and 𝑋 is an open set. (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ (𝐽 ∩ 𝒫 𝑍) ↔ (𝑌 ∈ 𝐽 ∧ (𝑋 ∩ 𝑌) = ∅)))
Theorem	clddisj 49395	Two ways of saying that two closed sets are disjoint, if 𝐽 is a topology and 𝑋 is a closed set. An alternative proof is similar to that of opndisj 49394 with elssuni 4882 replaced by the combination of cldss 23008 and eqid 2737. (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝑍 = (∪ 𝐽 ∖ 𝑋) → (𝑌 ∈ ((Clsd‘𝐽) ∩ 𝒫 𝑍) ↔ (𝑌 ∈ (Clsd‘𝐽) ∧ (𝑋 ∩ 𝑌) = ∅)))
21.49.10.2  Neighborhoods
Theorem	neircl 49396	Reverse closure of the neighborhood operation. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by Zhi Wang, 16-Sep-2024.)
⊢ (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝐽 ∈ Top)
Theorem	opnneilem 49397*	Lemma factoring out common proof steps of opnneil 49401 and opnneirv 49399. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))
Theorem	opnneir 49398*	If something is true for an open neighborhood, it must be true for a neighborhood. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) → ∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓))
Theorem	opnneirv 49399*	A variant of opnneir 49398 with different dummy variables. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ 𝐽 (𝑆 ⊆ 𝑥 ∧ 𝜓) → ∃𝑦 ∈ ((nei‘𝐽)‘𝑆)𝜒))
Theorem	opnneilv 49400*	The converse of opnneir 49398 with different dummy variables. Note that the second hypothesis could be generalized by adding 𝑦 ∈ 𝐽 to the antecedent. See the proof for details. Although 𝐽 ∈ Top might be redundant here (see neircl 49396), it is listed for explicitness. (Contributed by Zhi Wang, 31-Aug-2024.)
⊢ (𝜑 → 𝐽 ∈ Top)    &   ⊢ ((𝜑 ∧ 𝑦 ⊆ 𝑥) → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ((nei‘𝐽)‘𝑆)𝜓 → ∃𝑦 ∈ 𝐽 (𝑆 ⊆ 𝑦 ∧ 𝜒)))