P. 500 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 49901-50000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	imbi12d2a 49901	Variant of imbi12d2 49900. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝜓) → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
Theorem	imbi12d3 49902	Variant of imbi12d2 49900. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
Theorem	pm5.32rda 49903	Distribution of implication over biconditional (deduction form). Variant of pm5.32da 590. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜓)))
Theorem	pm5.32dar 49904	Reverse distribution of implication over biconditional (deduction form). (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜓 ∧ 𝜃)))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))
Theorem	exp12bd 49905	The import-export theorem (impexp 456) for biconditionals (deduction form). (Contributed by Zhi Wang, 3-Sep-2024.)
⊢ (𝜑 → (((𝜓 ∧ 𝜒) → 𝜃) ↔ ((𝜏 ∧ 𝜂) → 𝜁)))    ⇒   ⊢ (𝜑 → ((𝜓 → (𝜒 → 𝜃)) ↔ (𝜏 → (𝜂 → 𝜁))))
Theorem	mpbiran3d 49906	Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 24-Sep-2024.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    &   ⊢ ((𝜑 ∧ 𝜒) → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	mpbiran4d 49907	Equivalence with a conjunction one of whose conjuncts is a consequence of the other. Deduction form. (Contributed by Zhi Wang, 27-Sep-2024.)
⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃)))    &   ⊢ ((𝜑 ∧ 𝜃) → 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜃))
21.51.2  Predicate calculus with equality
21.51.2.1  Axiom scheme ax-5 (Distinctness)
Theorem	dtrucor3 49908*	An example of how ax-5 1943 without a distinct variable condition causes paradox in models of at least two objects. The hypothesis "dtrucor3.1" is provable from dtru 5405 in the ZF set theory. axc16nf 2298 and euae 2685 demonstrate that the violation of dtru 5405 leads to a model with only one object assuming its existence (ax-6 2000). The conclusion is also provable in the empty model ( see emptyal 1941). See also nf5 2316 and nf5i 2183 for the relation between unconditional ax-5 1943 and being not free. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦    &   ⊢ (𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)    ⇒   ⊢ ∀𝑥 𝑥 = 𝑦
21.51.3  ZF Set Theory - start with the Axiom of Extensionality
21.51.3.1  Restricted quantification
Theorem	ralbidb 49909*	Formula-building rule for restricted universal quantifier and additional condition (deduction form). See ralbidc 49910 for a more generalized form. (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 (𝜓 → 𝜃)))
Theorem	ralbidc 49910*	Formula-building rule for restricted universal quantifier and additional condition (deduction form). A variant of ralbidb 49909. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐵 ∧ 𝜓)))    &   ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 ∧ 𝜓)) → (𝜒 ↔ 𝜃)))    ⇒   ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜒 ↔ ∀𝑥 ∈ 𝐵 (𝜓 → 𝜃)))
Theorem	r19.41dv 49911*	A complex deduction form of r19.41v 3193. (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))
Theorem	rmotru 49912	Two ways of expressing "at most one" element. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof shortened by BJ, 23-Sep-2024.)
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃*𝑥 ∈ 𝐴 ⊤)
Theorem	reutru 49913	Two ways of expressing "exactly one" element. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)
Theorem	reutruALT 49914	Alternate proof of reutru 49913. (Contributed by Zhi Wang, 23-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)
Theorem	reueqbidva 49915*	Formula-building rule for restricted existential uniqueness quantifier. Deduction form. General version of reueqbidv 3402. (Contributed by Zhi Wang, 20-Nov-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒))
21.51.3.2  The universal class
Theorem	reuxfr1dd 49916*	Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Simplifies reuxfr1d 3708. (Contributed by Zhi Wang, 20-Sep-2025.)
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴)    &   ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐶 ∧ 𝑥 = 𝐴)) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒))
21.51.3.3  The empty set
Theorem	ssdisjd 49917	Subset preserves disjointness. Deduction form of ssdisj 4413. (Contributed by Zhi Wang, 7-Sep-2024.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝐵 ∩ 𝐶) = ∅)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅)
Theorem	ssdisjdr 49918	Subset preserves disjointness. Deduction form of ssdisj 4413. Alternatively this could be proved with ineqcom 4156 in tandem with ssdisjd 49917. (Contributed by Zhi Wang, 7-Sep-2024.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝐶 ∩ 𝐵) = ∅)    ⇒   ⊢ (𝜑 → (𝐶 ∩ 𝐴) = ∅)
Theorem	disjdifb 49919	Relative complement is anticommutative regarding intersection. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅
Theorem	predisj 49920	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) = ∅)
21.51.3.4  Unordered and ordered pairs
Theorem	vsn 49921	The singleton of the universal class is the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ {V} = ∅
Theorem	mosn 49922*	"At most one" element in a singleton. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
Theorem	mo0 49923*	"At most one" element in an empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)
Theorem	mosssn 49924*	"At most one" element in a subclass of a singleton. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
Theorem	mo0sn 49925*	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 49926*	Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦})
21.51.3.5  The union of a class
Theorem	unilbss 49927*	Superclass of the greatest lower bound. A dual statement of ssintub 4926. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴
21.51.3.6  Indexed union and intersection
Theorem	iuneq0 49928	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 49929	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 49930*	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 49931*	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 49932*	Indexed union of identical classes. (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)
Theorem	iineqconst2 49933*	Indexed intersection of identical classes. (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐶)
21.51.4  ZF Set Theory - add the Axiom of Replacement
21.51.4.1  Theorems requiring subset and intersection existence
Theorem	inpw 49934*	Two ways of expressing a collection of subsets as seen in df-ntr 23338, unimax 4905, and others. (Contributed by Zhi Wang, 27-Sep-2024.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵})
21.51.5  ZF Set Theory - add the Axiom of Power Sets
21.51.5.1  Ordered pair theorem
Theorem	opth1neg 49935	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 49936	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⟩))
21.51.5.2  Ordered-pair class abstractions (cont.)
Theorem	brab2dd 49937*	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 49938*	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 49939*	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)))
21.51.5.3  Relations
Theorem	iinxp 49940*	Indexed intersection of Cartesian products is the Cartesian product of indexed intersections. See also inxp 5809 and intxpd 49941. (Contributed by Zhi Wang, 30-Oct-2025.)
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) = (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶))
Theorem	intxpd 49941*	Intersection of Cartesian products is the Cartesian product of intersection of domains and ranges. See also inxp 5809 and iinxp 49940. (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 49942	Composition with a Cartesian product. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (𝐴 ∘ (𝐵 × 𝐶)) = (𝐵 × (𝐴 “ 𝐶))
Theorem	cosn 49943	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 49944	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 49945	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 49946	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))
21.51.5.4  Functions
Theorem	mof0 49947	There is at most one function into the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ ∃*𝑓 𝑓:𝐴⟶∅
Theorem	mof02 49948*	A variant of mof0 49947. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵)
Theorem	mof0ALT 49949*	Alternate proof of mof0 49947 with stronger requirements on distinct variables. Uses mo4 2592. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃*𝑓 𝑓:𝐴⟶∅
Theorem	eufsnlem 49950*	There is exactly one function into a singleton. For a simpler hypothesis, see eufsn 49951 assuming ax-rep 5232, or eufsn2 49952 assuming ax-pow 5327 and ax-un 7751. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → (𝐴 × {𝐵}) ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Theorem	eufsn 49951*	There is exactly one function into a singleton, assuming ax-rep 5232. See eufsn2 49952 for different axiom requirements. If existence is not needed, use mofsn 49953 or mofsn2 49954 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Theorem	eufsn2 49952*	There is exactly one function into a singleton, assuming ax-pow 5327 and ax-un 7751. Variant of eufsn 49951. If existence is not needed, use mofsn 49953 or mofsn2 49954 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Theorem	mofsn 49953*	There is at most one function into a singleton, with fewer axioms than eufsn 49951 and eufsn2 49952. See also mofsn2 49954. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐵 ∈ 𝑉 → ∃*𝑓 𝑓:𝐴⟶{𝐵})
Theorem	mofsn2 49954*	There is at most one function into a singleton. An unconditional variant of mofsn 49953, 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 49955*	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 49956*	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 49957*	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 49958	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 49959	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 49960	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 49961	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 49962*	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 49963	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 49964*	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 49965	Relation with function value. (Contributed by Zhi Wang, 25-Nov-2025.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑋𝐹(𝐹‘𝑋))
Theorem	xpco2 49966	Composition of a Cartesian product with a function. (Contributed by Zhi Wang, 25-Nov-2025.)
⊢ (𝐹:𝐴⟶𝐵 → ((𝐵 × 𝐶) ∘ 𝐹) = (𝐴 × 𝐶))
21.51.5.5  Operations
Theorem	ovsng 49967	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 49968	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 49969	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 49970	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	ovconstbrd 49971	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	ovconstbrn0d 49972	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	elovconstbrd 49973	Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐹𝐵))    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐵)
Theorem	ovmpt4d 49974*	Deduction version of ovmpt4g 7567. (This is the operation analogue of fvmpt2d 7007.) (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 49975*	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)
21.51.6  ZF Set Theory - add the Axiom of Union
21.51.6.1  Relations and functions (cont.)
Theorem	fonex 49976	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
21.51.6.2  First and second members of an ordered pair
Theorem	eloprab1st2nd 49977*	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)⟩)
21.51.6.3  Function transposition
Theorem	resinsnlem 49978	Lemma for resinsnALT 49980. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (𝜑 → (𝜒 ↔ ¬ 𝜓))    &   ⊢ (¬ 𝜑 → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ 𝜒)
Theorem	resinsn 49979	Restriction to the intersection with a singleton. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ ((𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ (dom 𝐹 ∩ 𝐴))
Theorem	resinsnALT 49980	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 49981*	Alternate definition of tpos. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ tpos 𝐹 = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))
Theorem	dftpos6 49982*	Alternate definition of tpos. The second half of the right hand side could apply ressn 6288 and become (𝐹 ↾ {∅}). (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ tpos 𝐹 = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ ({∅} × (𝐹 “ {∅})))
Theorem	dmtposss 49983	The domain of tpos 𝐹 is a subset. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})
Theorem	tposres0 49984	The transposition of a set restricted to the empty set is the set restricted to the empty set. See also ressn 6288 and dftpos6 49982 for an alternate proof. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅})
Theorem	tposresg 49985	The transposition restricted to a set. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ 𝑅) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))
Theorem	tposrescnv 49986*	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 8243. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ ◡𝑅) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥}))
Theorem	tposres2 49987	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 49988	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 49989	The transposition restricted to a relation. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (Rel 𝑅 → (tpos 𝐹 ↾ 𝑅) = tpos (𝐹 ↾ ◡𝑅))
Theorem	tposresxp 49990	The transposition restricted to a Cartesian product. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ (𝐴 × 𝐵)) = tpos (𝐹 ↾ (𝐵 × 𝐴))
Theorem	tposf1o 49991	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 49992	Swap an ordered pair. (Contributed by Zhi Wang, 5-Oct-2025.)
⊢ (𝑋tpos I 𝑌) = ⟨𝑌, 𝑋⟩
Theorem	tposidres 49993	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 49994	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 49995*	Two ways of expressing the swap function. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (Rel 𝑅 → (tpos I ↾ 𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ◡{𝑥}))
Theorem	tposideq2 49996*	Two ways of expressing the swap function. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ 𝑅 = (𝐴 × 𝐵)    ⇒   ⊢ (tpos I ↾ 𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ◡{𝑥})
21.51.6.4  Infinite Cartesian products
Theorem	ixpv 49997*	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}
21.51.6.5  Equinumerosity
Theorem	fvconst0ci 49998	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 49999	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 50000*	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 49999 assuming ax-un 7751 (see f1omoALT 50002). (Contributed by Zhi Wang, 19-Sep-2024.) (Proof shortened by SN, 24-Nov-2025.)
⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))    ⇒   ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))