P. 490 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48901-49000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	inlinecirc02plem 48901*	Lemma for inlinecirc02p 48902. (Contributed by AV, 7-May-2023.) (Revised by AV, 15-May-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝑄 = ((𝐴↑2) + (𝐵↑2))    &   ⊢ 𝐷 = (((𝑅↑2) · 𝑄) − (𝐶↑2))    &   ⊢ 𝐴 = ((𝑋‘2) − (𝑌‘2))    &   ⊢ 𝐵 = ((𝑌‘1) − (𝑋‘1))    &   ⊢ 𝐶 = (((𝑋‘2) · (𝑌‘1)) − ((𝑋‘1) · (𝑌‘2)))    ⇒   ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ 0 < 𝐷)) → ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ((( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) = {𝑎, 𝑏} ∧ 𝑎 ≠ 𝑏))
Theorem	inlinecirc02p 48902	Intersection of a line with a circle: A line passing through a point within a circle around the origin intersects the circle at exactly two different points. (Contributed by AV, 9-May-2023.) (Revised by AV, 16-May-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌)) ∈ (Pairsproper‘𝑃))
Theorem	inlinecirc02preu 48903*	Intersection of a line with a circle: A line passing through a point within a circle around the origin intersects the circle at exactly two different points, expressed with restricted uniqueness (and without the definition of proper pairs). (Contributed by AV, 16-May-2023.)
⊢ 𝐼 = {1, 2}    &   ⊢ 𝐸 = (ℝ^‘𝐼)    &   ⊢ 𝑃 = (ℝ ↑m 𝐼)    &   ⊢ 𝑆 = (Sphere‘𝐸)    &   ⊢ 0 = (𝐼 × {0})    &   ⊢ 𝐿 = (LineM‘𝐸)    &   ⊢ 𝐷 = (dist‘𝐸)    ⇒   ⊢ (((𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ≠ 𝑌) ∧ (𝑅 ∈ ℝ+ ∧ (𝑋𝐷 0 ) < 𝑅)) → ∃!𝑝 ∈ 𝒫 𝑃((♯‘𝑝) = 2 ∧ 𝑝 = (( 0 𝑆𝑅) ∩ (𝑋𝐿𝑌))))
21.49  Mathbox for Zhi Wang
21.49.1  Propositional calculus
Theorem	pm4.71da 48904	Deduction converting a biconditional to a biconditional with conjunction. Variant of pm4.71d 561. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ (𝜓 ∧ 𝜒)))
Theorem	logic1 48905	Distribution of implication over biconditional with replacement (deduction form). (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜓 → (𝜃 ↔ 𝜏)))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
Theorem	logic1a 48906	Variant of logic1 48905. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ ((𝜑 ∧ 𝜓) → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
Theorem	logic2 48907	Variant of logic1 48905. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → ((𝜓 ∧ 𝜒) → (𝜃 ↔ 𝜏)))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜃) ↔ (𝜒 → 𝜏)))
Theorem	pm5.32dav 48908	Distribution of implication over biconditional (deduction form). Variant of pm5.32da 579. (Contributed by Zhi Wang, 30-Aug-2024.)
⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → ((𝜒 ∧ 𝜓) ↔ (𝜃 ∧ 𝜓)))
Theorem	pm5.32dra 48909	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 48910	The import-export theorem (impexp 450) 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 48911	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 48912	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))
21.49.2  Predicate calculus with equality
21.49.2.1  Axiom scheme ax-5 (Distinctness)
Theorem	dtrucor3 48913*	An example of how ax-5 1911 without a distinct variable condition causes paradox in models of at least two objects. The hypothesis "dtrucor3.1" is provable from dtru 5383 in the ZF set theory. axc16nf 2268 and euae 2657 demonstrate that the violation of dtru 5383 leads to a model with only one object assuming its existence (ax-6 1968). The conclusion is also provable in the empty model ( see emptyal 1909). See also nf5 2286 and nf5i 2151 for the relation between unconditional ax-5 1911 and being not free. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ ¬ ∀𝑥 𝑥 = 𝑦    &   ⊢ (𝑥 = 𝑦 → ∀𝑥 𝑥 = 𝑦)    ⇒   ⊢ ∀𝑥 𝑥 = 𝑦
21.49.3  ZF Set Theory - start with the Axiom of Extensionality
21.49.3.1  Restricted quantification
Theorem	ralbidb 48914*	Formula-building rule for restricted universal quantifier and additional condition (deduction form). See ralbidc 48915 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 48915*	Formula-building rule for restricted universal quantifier and additional condition (deduction form). A variant of ralbidb 48914. (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 48916*	A complex deduction form of r19.41v 3164. (Contributed by Zhi Wang, 6-Sep-2024.)
⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝜓)    ⇒   ⊢ ((𝜑 ∧ 𝜒) → ∃𝑥 ∈ 𝐴 (𝜓 ∧ 𝜒))
Theorem	rmotru 48917	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 48918	Two ways of expressing "exactly one" element. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)
Theorem	reutruALT 48919	Alternate proof of reutru 48918. (Contributed by Zhi Wang, 23-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥 ∈ 𝐴 ⊤)
Theorem	reueqbidva 48920*	Formula-building rule for restricted existential uniqueness quantifier. Deduction form. General version of reueqbidv 3386. (Contributed by Zhi Wang, 20-Nov-2025.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐵 𝜒))
21.49.3.2  The universal class
Theorem	reuxfr1dd 48921*	Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Simplifies reuxfr1d 3706. (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))
21.49.3.3  The empty set
Theorem	ssdisjd 48922	Subset preserves disjointness. Deduction form of ssdisj 4411. (Contributed by Zhi Wang, 7-Sep-2024.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝐵 ∩ 𝐶) = ∅)    ⇒   ⊢ (𝜑 → (𝐴 ∩ 𝐶) = ∅)
Theorem	ssdisjdr 48923	Subset preserves disjointness. Deduction form of ssdisj 4411. Alternatively this could be proved with ineqcom 4161 in tandem with ssdisjd 48922. (Contributed by Zhi Wang, 7-Sep-2024.)
⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → (𝐶 ∩ 𝐵) = ∅)    ⇒   ⊢ (𝜑 → (𝐶 ∩ 𝐴) = ∅)
Theorem	disjdifb 48924	Relative complement is anticommutative regarding intersection. (Contributed by Zhi Wang, 5-Sep-2024.)
⊢ ((𝐴 ∖ 𝐵) ∩ (𝐵 ∖ 𝐴)) = ∅
Theorem	predisj 48925	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 48926	The singleton of the universal class is the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ {V} = ∅
Theorem	mosn 48927*	"At most one" element in a singleton. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐴 = {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
Theorem	mo0 48928*	"At most one" element in an empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐴 = ∅ → ∃*𝑥 𝑥 ∈ 𝐴)
Theorem	mosssn 48929*	"At most one" element in a subclass of a singleton. (Contributed by Zhi Wang, 23-Sep-2024.)
⊢ (𝐴 ⊆ {𝐵} → ∃*𝑥 𝑥 ∈ 𝐴)
Theorem	mo0sn 48930*	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 48931*	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 48932*	Superclass of the greatest lower bound. A dual statement of ssintub 4918. (Contributed by Zhi Wang, 29-Sep-2024.)
⊢ ∪ {𝑥 ∈ 𝐵 ∣ 𝑥 ⊆ 𝐴} ⊆ 𝐴
21.49.3.6  Indexed union and intersection
Theorem	iuneq0 48933	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 48934	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 48935*	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 48936*	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 48937*	Indexed union of identical classes. (Contributed by Zhi Wang, 6-Nov-2025.)
⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐶)
Theorem	iineqconst2 48938*	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 48939*	Two ways of expressing a collection of subsets as seen in df-ntr 22945, unimax 4897, 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 48940	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 48941	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 48942*	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 48943*	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 48944*	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 48945*	Indexed intersection of Cartesian products is the Cartesian product of indexed intersections. See also inxp 5778 and intxp 48946. (Contributed by Zhi Wang, 30-Oct-2025.)
⊢ (𝐴 ≠ ∅ → ∩ 𝑥 ∈ 𝐴 (𝐵 × 𝐶) = (∩ 𝑥 ∈ 𝐴 𝐵 × ∩ 𝑥 ∈ 𝐴 𝐶))
Theorem	intxp 48946*	Intersection of Cartesian products is the Cartesian product of intersection of domains and ranges. See also inxp 5778 and iinxp 48945. (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 48947	Composition with a Cartesian product. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (𝐴 ∘ (𝐵 × 𝐶)) = (𝐵 × (𝐴 “ 𝐶))
Theorem	cosn 48948	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 48949	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 48950	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 48951	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 48952	There is at most one function into the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ ∃*𝑓 𝑓:𝐴⟶∅
Theorem	mof02 48953*	A variant of mof0 48952. (Contributed by Zhi Wang, 20-Sep-2024.)
⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵)
Theorem	mof0ALT 48954*	Alternate proof of mof0 48952 with stronger requirements on distinct variables. Uses mo4 2563. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ∃*𝑓 𝑓:𝐴⟶∅
Theorem	eufsnlem 48955*	There is exactly one function into a singleton. For a simpler hypothesis, see eufsn 48956 assuming ax-rep 5221, or eufsn2 48957 assuming ax-pow 5307 and ax-un 7677. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → (𝐴 × {𝐵}) ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Theorem	eufsn 48956*	There is exactly one function into a singleton, assuming ax-rep 5221. See eufsn2 48957 for different axiom requirements. If existence is not needed, use mofsn 48958 or mofsn2 48959 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Theorem	eufsn2 48957*	There is exactly one function into a singleton, assuming ax-pow 5307 and ax-un 7677. Variant of eufsn 48956. If existence is not needed, use mofsn 48958 or mofsn2 48959 for fewer axiom assumptions. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃!𝑓 𝑓:𝐴⟶{𝐵})
Theorem	mofsn 48958*	There is at most one function into a singleton, with fewer axioms than eufsn 48956 and eufsn2 48957. See also mofsn2 48959. (Contributed by Zhi Wang, 19-Sep-2024.)
⊢ (𝐵 ∈ 𝑉 → ∃*𝑓 𝑓:𝐴⟶{𝐵})
Theorem	mofsn2 48959*	There is at most one function into a singleton. An unconditional variant of mofsn 48958, 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 48960*	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 48961*	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 48962*	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 48963	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 48964	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 48965	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 48966	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 48967*	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 48968	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 48969*	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 48970	Relation with function value. (Contributed by Zhi Wang, 25-Nov-2025.)
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → 𝑋𝐹(𝐹‘𝑋))
Theorem	xpco2 48971	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 48972	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 48973	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 48974	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 48975	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 48976	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 48977	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 48978	Two ways of expressing 𝐴𝑅𝐵. (Contributed by Zhi Wang, 18-Sep-2024.)
⊢ (𝜑 → 𝐹 = (𝑅 × {𝑌}))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐴𝐹𝐵))    ⇒   ⊢ (𝜑 → 𝐴𝑅𝐵)
Theorem	ovmpt4d 48979*	Deduction version of ovmpt4g 7502. (This is the operation analogue of fvmpt2d 6951.) (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 48980*	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 48981	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 48982*	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 48983*	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 48984*	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 48985	Lemma for resinsnALT 48987. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (𝜑 → (𝜒 ↔ ¬ 𝜓))    &   ⊢ (¬ 𝜑 → 𝜒)    ⇒   ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ 𝜒)
Theorem	resinsn 48986	Restriction to the intersection with a singleton. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ ((𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ (dom 𝐹 ∩ 𝐴))
Theorem	resinsnALT 48987	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 48988*	Alternate definition of tpos. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ tpos 𝐹 = (𝐹 ∘ ((𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥}) ∪ {⟨∅, ∅⟩}))
Theorem	dftpos6 48989*	Alternate definition of tpos. The second half of the right hand side could apply ressn 6240 and become (𝐹 ↾ {∅}) (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ tpos 𝐹 = ((𝐹 ∘ (𝑥 ∈ ◡dom 𝐹 ↦ ∪ ◡{𝑥})) ∪ ({∅} × (𝐹 “ {∅})))
Theorem	dmtposss 48990	The domain of tpos 𝐹 is a subset. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ dom tpos 𝐹 ⊆ ((V × V) ∪ {∅})
Theorem	tposres0 48991	The transposition of a set restricted to the empty set is the set restricted to the empty set. See also ressn 6240 and dftpos6 48989 for an alternate proof. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ {∅}) = (𝐹 ↾ {∅})
Theorem	tposresg 48992	The transposition restricted to a set. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ 𝑅) = ((tpos 𝐹 ↾ ◡◡𝑅) ∪ (𝐹 ↾ (𝑅 ∩ {∅})))
Theorem	tposrescnv 48993*	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 8165. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ ◡𝑅) = (𝐹 ∘ (𝑥 ∈ ◡dom (𝐹 ↾ 𝑅) ↦ ∪ ◡{𝑥}))
Theorem	tposres2 48994	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 48995	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 48996	The transposition restricted to a relation. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (Rel 𝑅 → (tpos 𝐹 ↾ 𝑅) = tpos (𝐹 ↾ ◡𝑅))
Theorem	tposresxp 48997	The transposition restricted to a Cartesian product. (Contributed by Zhi Wang, 6-Oct-2025.)
⊢ (tpos 𝐹 ↾ (𝐴 × 𝐵)) = tpos (𝐹 ↾ (𝐵 × 𝐴))
Theorem	tposf1o 48998	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 48999	Swap an ordered pair. (Contributed by Zhi Wang, 5-Oct-2025.)
⊢ (𝑋tpos I 𝑌) = ⟨𝑌, 𝑋⟩
Theorem	tposidres 49000	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⟩)