P. 497 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 49601-49700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pm5.32dra 49601	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 49602	The import-export theorem (impexp 455) 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 49603	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 49604	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))
21.50.2  Predicate calculus with equality
21.50.2.1  Axiom scheme ax-5 (Distinctness)
Theorem	dtrucor3 49605*	An example of how ax-5 1940 without a distinct variable condition causes paradox in models of at least two objects. The hypothesis "dtrucor3.1" is provable from dtru 5418 in the ZF set theory. axc16nf 2299 and euae 2687 demonstrate that the violation of dtru 5418 leads to a model with only one object assuming its existence (ax-6 1997). The conclusion is also provable in the empty model ( see emptyal 1938). See also nf5 2317 and nf5i 2181 for the relation between unconditional ax-5 1940 and being not free. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦    &   ⊢ (𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)    ⇒   ⊢ ∀𝑥 𝑥 = 𝑦
21.50.3  ZF Set Theory - start with the Axiom of Extensionality
21.50.3.1  Restricted quantification
Theorem	ralbidb 49606*	Formula-building rule for restricted universal quantifier and additional condition (deduction form). See ralbidc 49607 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 49607*	Formula-building rule for restricted universal quantifier and additional condition (deduction form). A variant of ralbidb 49606. (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 49608*	A complex deduction form of r19.41v 3195. (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))
Theorem	rmotru 49609	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 49610	Two ways of expressing "exactly one" element. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)
Theorem	reutruALT 49611	Alternate proof of reutru 49610. (Contributed by Zhi Wang, 23-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)
Theorem	reueqbidva 49612*	Formula-building rule for restricted existential uniqueness quantifier. Deduction form. General version of reueqbidv 3405. (Contributed by Zhi Wang, 20-Nov-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒))
21.50.3.2  The universal class
Theorem	reuxfr1dd 49613*	Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Simplifies reuxfr1d 3713. (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))
21.50.3.3  The empty set
Theorem	ssdisjd 49614	Subset preserves disjointness. Deduction form of ssdisj 4420. (Contributed by Zhi Wang, 7-Sep-2024.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝐵 ∩ 𝐶) = ∅)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅)
Theorem	ssdisjdr 49615	Subset preserves disjointness. Deduction form of ssdisj 4420. Alternatively this could be proved with ineqcom 4163 in tandem with ssdisjd 49614. (Contributed by Zhi Wang, 7-Sep-2024.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝐶 ∩ 𝐵) = ∅)    ⇒   ⊢ (𝜑 → (𝐶 ∩ 𝐴) = ∅)
Theorem	disjdifb 49616	Relative complement is anticommutative regarding intersection. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅
Theorem	predisj 49617	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) = ∅)
21.50.3.4  Unordered and ordered pairs
Theorem	vsn 49618	The singleton of the universal class is the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ {V} = ∅
Theorem	mosn 49619*	"At most one" element in a singleton. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
Theorem	mo0 49620*	"At most one" element in an empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)
Theorem	mosssn 49621*	"At most one" element in a subclass of a singleton. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
Theorem	mo0sn 49622*	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 49623*	Two ways of expressing "at most one" element in a class. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝐴 ⊆ {𝑦})
21.50.3.5  The union of a class
Theorem	unilbss 49624*	Superclass of the greatest lower bound. A dual statement of ssintub 4931. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴
21.50.3.6  Indexed union and intersection
Theorem	iuneq0 49625	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 49626	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 49627*	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 49628*	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 49629*	Indexed union of identical classes. (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)
Theorem	iineqconst2 49630*	Indexed intersection of identical classes. (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 = 𝐶)
21.50.4  ZF Set Theory - add the Axiom of Replacement
21.50.4.1  Theorems requiring subset and intersection existence
Theorem	inpw 49631*	Two ways of expressing a collection of subsets as seen in df-ntr 23186, unimax 4910, and others. (Contributed by Zhi Wang, 27-Sep-2024.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝑥 ⊆ 𝐵})
21.50.5  ZF Set Theory - add the Axiom of Power Sets
21.50.5.1  Ordered pair theorem
Theorem	opth1neg 49632	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 49633	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⟩))
21.50.5.2  Ordered-pair class abstractions (cont.)
Theorem	brab2dd 49634*	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 49635*	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 49636*	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)))
21.50.5.3  Relations
Theorem	iinxp 49637*	Indexed intersection of Cartesian products is the Cartesian product of indexed intersections. See also inxp 5818 and intxp 49638. (Contributed by Zhi Wang, 30-Oct-2025.)
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) = (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶))
Theorem	intxp 49638*	Intersection of Cartesian products is the Cartesian product of intersection of domains and ranges. See also inxp 5818 and iinxp 49637. (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 49639	Composition with a Cartesian product. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (𝐴 ∘ (𝐵 × 𝐶)) = (𝐵 × (𝐴 “ 𝐶))
Theorem	cosn 49640	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 49641	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 49642	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 49643	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))
21.50.5.4  Functions
Theorem	mof0 49644	There is at most one function into the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ ∃*𝑓 𝑓:𝐴⟶∅
Theorem	mof02 49645*	A variant of mof0 49644. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵)
Theorem	mof0ALT 49646*	Alternate proof of mof0 49644 with stronger requirements on distinct variables. Uses mo4 2594. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃*𝑓 𝑓:𝐴⟶∅
Theorem	eufsnlem 49647*	There is exactly one function into a singleton. For a simpler hypothesis, see eufsn 49648 assuming ax-rep 5238, or eufsn2 49649 assuming ax-pow 5336 and ax-un 7732. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → (𝐴 × {𝐵}) ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Theorem	eufsn 49648*	There is exactly one function into a singleton, assuming ax-rep 5238. See eufsn2 49649 for different axiom requirements. If existence is not needed, use mofsn 49650 or mofsn2 49651 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Theorem	eufsn2 49649*	There is exactly one function into a singleton, assuming ax-pow 5336 and ax-un 7732. Variant of eufsn 49648. If existence is not needed, use mofsn 49650 or mofsn2 49651 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Theorem	mofsn 49650*	There is at most one function into a singleton, with fewer axioms than eufsn 49648 and eufsn2 49649. See also mofsn2 49651. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐵 ∈ 𝑉 → ∃*𝑓 𝑓:𝐴⟶{𝐵})
Theorem	mofsn2 49651*	There is at most one function into a singleton. An unconditional variant of mofsn 49650, 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 49652*	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 49653*	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 49654*	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 49655	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 49656	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 49657	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 49658	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 49659*	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 49660	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 49661*	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 49662	Relation with function value. (Contributed by Zhi Wang, 25-Nov-2025.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑋𝐹(𝐹‘𝑋))
Theorem	xpco2 49663	Composition of a Cartesian product with a function. (Contributed by Zhi Wang, 25-Nov-2025.)
⊢ (𝐹:𝐴⟶𝐵 → ((𝐵 × 𝐶) ∘ 𝐹) = (𝐴 × 𝐶))
21.50.5.5  Operations
Theorem	ovsng 49664	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 49665	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 49666	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 49667	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 49668	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 49669	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 49670	Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐹𝐵))    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐵)
Theorem	ovmpt4d 49671*	Deduction version of ovmpt4g 7557. (This is the operation analogue of fvmpt2d 7003.) (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 49672*	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)
21.50.6  ZF Set Theory - add the Axiom of Union
21.50.6.1  Relations and functions (cont.)
Theorem	fonex 49673	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
21.50.6.2  First and second members of an ordered pair
Theorem	eloprab1st2nd 49674*	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)⟩)
21.50.6.3  Operations in maps-to notation (continued)
Theorem	fmpodg 49675*	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 49676*	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)
21.50.6.4  Function transposition
Theorem	resinsnlem 49677	Lemma for resinsnALT 49679. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (𝜑 → (𝜒 ↔ ¬ 𝜓))    &   ⊢ (¬ 𝜑 → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ 𝜒)
Theorem	resinsn 49678	Restriction to the intersection with a singleton. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ ((𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ (dom 𝐹 ∩ 𝐴))
Theorem	resinsnALT 49679	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 49680*	Alternate definition of tpos. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ tpos 𝐹 = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))
Theorem	dftpos6 49681*	Alternate definition of tpos. The second half of the right hand side could apply ressn 6286 and become (𝐹 ↾ {∅}). (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ tpos 𝐹 = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ ({∅} × (𝐹 “ {∅})))
Theorem	dmtposss 49682	The domain of tpos 𝐹 is a subset. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})
Theorem	tposres0 49683	The transposition of a set restricted to the empty set is the set restricted to the empty set. See also ressn 6286 and dftpos6 49681 for an alternate proof. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅})
Theorem	tposresg 49684	The transposition restricted to a set. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ 𝑅) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))
Theorem	tposrescnv 49685*	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 8218. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ ◡𝑅) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥}))
Theorem	tposres2 49686	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 49687	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 49688	The transposition restricted to a relation. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (Rel 𝑅 → (tpos 𝐹 ↾ 𝑅) = tpos (𝐹 ↾ ◡𝑅))
Theorem	tposresxp 49689	The transposition restricted to a Cartesian product. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ (𝐴 × 𝐵)) = tpos (𝐹 ↾ (𝐵 × 𝐴))
Theorem	tposf1o 49690	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 49691	Swap an ordered pair. (Contributed by Zhi Wang, 5-Oct-2025.)
⊢ (𝑋tpos I 𝑌) = ⟨𝑌, 𝑋⟩
Theorem	tposidres 49692	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 49693	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 49694*	Two ways of expressing the swap function. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (Rel 𝑅 → (tpos I ↾ 𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ◡{𝑥}))
Theorem	tposideq2 49695*	Two ways of expressing the swap function. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ 𝑅 = (𝐴 × 𝐵)    ⇒   ⊢ (tpos I ↾ 𝑅) = (𝑥 ∈ 𝑅 ↦ ∪ ◡{𝑥})
21.50.6.5  Infinite Cartesian products
Theorem	ixpv 49696*	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}
21.50.6.6  Equinumerosity
Theorem	fvconst0ci 49697	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 49698	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 49699*	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 49698 assuming ax-un 7732 (see f1omoALT 49701). (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 49700*	Obsolete version of f1omo 49699 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))