P. 56 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 5501-5600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xpindi 5501	Distributive law for Cartesian product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
⊢ (𝐴 × (𝐵 ∩ 𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))
Theorem	xpindir 5502	Distributive law for Cartesian product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
⊢ ((𝐴 ∩ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶))
Theorem	xpiindi 5503*	Distributive law for Cartesian product over indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (𝐴 ≠ ∅ → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
Theorem	xpriindi 5504*	Distributive law for Cartesian product over relativized indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
⊢ (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
Theorem	eliunxp 5505*	Membership in a union of Cartesian products. Analogue of elxp 5378 for nonconstant 𝐵(𝑥). (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ (𝐶 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
Theorem	opeliunxp2 5506*	Membership in a union of Cartesian products. (Contributed by Mario Carneiro, 14-Feb-2015.)
⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸)    ⇒   ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))
Theorem	raliunxp 5507*	Write a double restricted quantification as one universal quantifier. In this version of ralxp 5509, 𝐵(𝑦) is not assumed to be constant. (Contributed by Mario Carneiro, 29-Dec-2014.)
⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓)
Theorem	rexiunxp 5508*	Write a double restricted quantification as one universal quantifier. In this version of rexxp 5510, 𝐵(𝑦) is not assumed to be constant. (Contributed by Mario Carneiro, 14-Feb-2015.)
⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑦} × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)
Theorem	ralxp 5509*	Universal quantification restricted to a Cartesian product is equivalent to a double restricted quantification. The hypothesis specifies an implicit substitution. (Contributed by NM, 7-Feb-2004.) (Revised by Mario Carneiro, 29-Dec-2014.)
⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓)
Theorem	rexxp 5510*	Existential quantification restricted to a Cartesian product is equivalent to a double restricted quantification. (Contributed by NM, 11-Nov-1995.) (Revised by Mario Carneiro, 14-Feb-2015.)
⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)
Theorem	exopxfr 5511*	Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ (V × V)𝜑 ↔ ∃𝑦∃𝑧𝜓)
Theorem	exopxfr2 5512*	Transfer ordered-pair existence from/to single variable existence. (Contributed by NM, 26-Feb-2014.)
⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (Rel 𝐴 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑧(⟨𝑦, 𝑧⟩ ∈ 𝐴 ∧ 𝜓)))
Theorem	djussxp 5513*	Disjoint union is a subset of a Cartesian product. (Contributed by Stefan O'Rear, 21-Nov-2014.)
⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ⊆ (𝐴 × V)
Theorem	ralxpf 5514*	Version of ralxp 5509 with bound-variable hypotheses. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∀𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐵 𝜓)
Theorem	rexxpf 5515*	Version of rexxp 5510 with bound-variable hypotheses. (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑧𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑 ↔ 𝜓))    ⇒   ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝜓)
Theorem	iunxpf 5516*	Indexed union on a Cartesian product equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.)
⊢ Ⅎ𝑦𝐶    &   ⊢ Ⅎ𝑧𝐶    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)    ⇒   ⊢ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷
Theorem	opabbi2dv 5517*	Deduce equality of a relation and an ordered-pair class builder. Compare abbi2dv 2897. (Contributed by NM, 24-Feb-2014.)
⊢ Rel 𝐴    &   ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ 𝜓))    ⇒   ⊢ (𝜑 → 𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
Theorem	relop 5518*	A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.) (Avoid depending on this detail.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (Rel ⟨𝐴, 𝐵⟩ ↔ ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦}))
Theorem	ideqg 5519	For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐵 ∈ 𝑉 → (𝐴 I 𝐵 ↔ 𝐴 = 𝐵))
Theorem	ideq 5520	For sets, the identity relation is the same as equality. (Contributed by NM, 13-Aug-1995.)
⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴 I 𝐵 ↔ 𝐴 = 𝐵)
Theorem	ididg 5521	A set is identical to itself. (Contributed by NM, 28-May-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 I 𝐴)
Theorem	issetid 5522	Two ways of expressing set existence. (Contributed by NM, 16-Feb-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ (𝐴 ∈ V ↔ 𝐴 I 𝐴)
Theorem	coss1 5523	Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.)
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶))
Theorem	coss2 5524	Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.)
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∘ 𝐴) ⊆ (𝐶 ∘ 𝐵))
Theorem	coeq1 5525	Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
Theorem	coeq2 5526	Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.)
⊢ (𝐴 = 𝐵 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵))
Theorem	coeq1i 5527	Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)
Theorem	coeq2i 5528	Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)
Theorem	coeq1d 5529	Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
Theorem	coeq2d 5530	Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵))
Theorem	coeq12i 5531	Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐶 = 𝐷    ⇒   ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷)
Theorem	coeq12d 5532	Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷))
Theorem	nfco 5533	Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥(𝐴 ∘ 𝐵)
Theorem	brcog 5534*	Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)))
Theorem	opelco2g 5535*	Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐷 ∧ ⟨𝑥, 𝐵⟩ ∈ 𝐶)))
Theorem	brcogw 5536	Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝑍) ∧ (𝐴𝐷𝑋 ∧ 𝑋𝐶𝐵)) → 𝐴(𝐶 ∘ 𝐷)𝐵)
Theorem	eqbrrdva 5537*	Deduction from extensionality principle for relations, given an equivalence only on the relation domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)
⊢ (𝜑 → 𝐴 ⊆ (𝐶 × 𝐷))    &   ⊢ (𝜑 → 𝐵 ⊆ (𝐶 × 𝐷))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))    ⇒   ⊢ (𝜑 → 𝐴 = 𝐵)
Theorem	brco 5538*	Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))
Theorem	opelco 5539*	Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))
Theorem	cnvss 5540	Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Kyle Wyonch, 27-Apr-2021.)
⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)
Theorem	cnveq 5541	Equality theorem for converse relation. (Contributed by NM, 13-Aug-1995.)
⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵)
Theorem	cnveqi 5542	Equality inference for converse relation. (Contributed by NM, 23-Dec-2008.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ◡𝐴 = ◡𝐵
Theorem	cnveqd 5543	Equality deduction for converse relation. (Contributed by NM, 6-Dec-2013.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ◡𝐴 = ◡𝐵)
Theorem	elcnv 5544*	Membership in a converse relation. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)
⊢ (𝐴 ∈ ◡𝑅 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ 𝑦𝑅𝑥))
Theorem	elcnv2 5545*	Membership in a converse relation. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)
⊢ (𝐴 ∈ ◡𝑅 ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑦, 𝑥⟩ ∈ 𝑅))
Theorem	nfcnv 5546	Bound-variable hypothesis builder for converse relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥◡𝐴
Theorem	brcnvg 5547	The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 10-Oct-2005.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴))
Theorem	opelcnvg 5548	Ordered-pair membership in converse relation. (Contributed by NM, 13-May-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ ◡𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅))
Theorem	opelcnv 5549	Ordered-pair membership in converse relation. (Contributed by NM, 13-Aug-1995.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ ∈ ◡𝑅 ↔ ⟨𝐵, 𝐴⟩ ∈ 𝑅)
Theorem	brcnv 5550	The converse of a binary relation swaps arguments. Theorem 11 of [Suppes] p. 61. (Contributed by NM, 13-Aug-1995.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴◡𝑅𝐵 ↔ 𝐵𝑅𝐴)
Theorem	csbcnv 5551	Move class substitution in and out of the converse of a relation. Version of csbcnvgALT 5552 without a sethood antecedent but depending on more axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (Revised by NM, 23-Aug-2018.)
⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹
Theorem	csbcnvgALT 5552	Move class substitution in and out of the converse of a relation. Version of csbcnv 5551 with a sethood antecedent but depending on fewer axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹)
Theorem	cnvco 5553	Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ◡(𝐴 ∘ 𝐵) = (◡𝐵 ∘ ◡𝐴)
Theorem	cnvuni 5554*	The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004.)
⊢ ◡∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ◡𝑥
Theorem	dfdm3 5555*	Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴}
Theorem	dfrn2 5556*	Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)
⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Theorem	dfrn3 5557*	Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)
⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}
Theorem	elrn2g 5558*	Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥⟨𝑥, 𝐴⟩ ∈ 𝐵))
Theorem	elrng 5559*	Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))
Theorem	ssrelrn 5560*	If a relation is a subset of a cartesian product, then for each element of the range of the relation there is an element of the first set of the cartesian product which is related to the element of the range by the relation. (Contributed by AV, 24-Oct-2020.)
⊢ ((𝑅 ⊆ (𝐴 × 𝐵) ∧ 𝑌 ∈ ran 𝑅) → ∃𝑎 ∈ 𝐴 𝑎𝑅𝑌)
Theorem	dfdm4 5561	Alternate definition of domain. (Contributed by NM, 28-Dec-1996.)
⊢ dom 𝐴 = ran ◡𝐴
Theorem	dfdmf 5562*	Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑦𝐴    ⇒   ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
Theorem	csbdm 5563	Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.) (Revised by NM, 24-Aug-2018.)
⊢ ⦋𝐴 / 𝑥⦌dom 𝐵 = dom ⦋𝐴 / 𝑥⦌𝐵
Theorem	eldmg 5564*	Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦))
Theorem	eldm2g 5565*	Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 9-Jul-2014.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐵))
Theorem	eldm 5566*	Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 2-Apr-2004.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)
Theorem	eldm2 5567*	Membership in a domain. Theorem 4 of [Suppes] p. 59. (Contributed by NM, 1-Aug-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ dom 𝐵 ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ 𝐵)
Theorem	dmss 5568	Subset theorem for domain. (Contributed by NM, 11-Aug-1994.)
⊢ (𝐴 ⊆ 𝐵 → dom 𝐴 ⊆ dom 𝐵)
Theorem	dmeq 5569	Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
Theorem	dmeqi 5570	Equality inference for domain. (Contributed by NM, 4-Mar-2004.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ dom 𝐴 = dom 𝐵
Theorem	dmeqd 5571	Equality deduction for domain. (Contributed by NM, 4-Mar-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → dom 𝐴 = dom 𝐵)
Theorem	opeldmd 5572	Membership of first of an ordered pair in a domain. Deduction version of opeldm 5573. (Contributed by AV, 11-Mar-2021.)
⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    ⇒   ⊢ (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ dom 𝐶))
Theorem	opeldm 5573	Membership of first of an ordered pair in a domain. (Contributed by NM, 30-Jul-1995.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (⟨𝐴, 𝐵⟩ ∈ 𝐶 → 𝐴 ∈ dom 𝐶)
Theorem	breldm 5574	Membership of first of a binary relation in a domain. (Contributed by NM, 30-Jul-1995.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅)
Theorem	breldmg 5575	Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)
⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)
Theorem	dmun 5576	The domain of a union is the union of domains. Exercise 56(a) of [Enderton] p. 65. (Contributed by NM, 12-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ dom (𝐴 ∪ 𝐵) = (dom 𝐴 ∪ dom 𝐵)
Theorem	dmin 5577	The domain of an intersection belong to the intersection of domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.)
⊢ dom (𝐴 ∩ 𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵)
Theorem	dmiun 5578	The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.)
⊢ dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 dom 𝐵
Theorem	dmuni 5579*	The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by NM, 3-Feb-2004.)
⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥
Theorem	dmopab 5580*	The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
⊢ dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑}
Theorem	dmopabss 5581*	Upper bound for the domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
⊢ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
Theorem	dmopab3 5582*	The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
⊢ (∀𝑥 ∈ 𝐴 ∃𝑦𝜑 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} = 𝐴)
Theorem	opabssxpd 5583*	An ordered-pair class abstraction is a subset of a Cartesian product. Formerly part of proof for opabex2 7506. (Contributed by AV, 26-Nov-2021.)
⊢ ((𝜑 ∧ 𝜓) → 𝑥 ∈ 𝐴)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑦 ∈ 𝐵)    ⇒   ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
Theorem	dm0 5584	The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ dom ∅ = ∅
Theorem	dmi 5585	The domain of the identity relation is the universe. (Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ dom I = V
Theorem	dmv 5586	The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.)
⊢ dom V = V
Theorem	dm0rn0 5587	An empty domain is equivalent to an empty range. (Contributed by NM, 21-May-1998.)
⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅)
Theorem	reldm0 5588	A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)
⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
Theorem	dmxp 5589	The domain of a Cartesian product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
Theorem	dmxpid 5590	The domain of a square Cartesian product. (Contributed by NM, 28-Jul-1995.)
⊢ dom (𝐴 × 𝐴) = 𝐴
Theorem	dmxpin 5591	The domain of the intersection of two square Cartesian products. Unlike dmin 5577, equality holds. (Contributed by NM, 29-Jan-2008.)
⊢ dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴 ∩ 𝐵)
Theorem	xpid11 5592	The Cartesian product of a class with itself is one-to-one. (Contributed by NM, 5-Nov-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵)
Theorem	dmcnvcnv 5593	The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 5837). (Contributed by NM, 8-Apr-2007.)
⊢ dom ◡◡𝐴 = dom 𝐴
Theorem	rncnvcnv 5594	The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)
⊢ ran ◡◡𝐴 = ran 𝐴
Theorem	elreldm 5595	The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)
⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴)
Theorem	rneq 5596	Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
Theorem	rneqi 5597	Equality inference for range. (Contributed by NM, 4-Mar-2004.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ ran 𝐴 = ran 𝐵
Theorem	rneqd 5598	Equality deduction for range. (Contributed by NM, 4-Mar-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ran 𝐴 = ran 𝐵)
Theorem	rnss 5599	Subset theorem for range. (Contributed by NM, 22-Mar-1998.)
⊢ (𝐴 ⊆ 𝐵 → ran 𝐴 ⊆ ran 𝐵)
Theorem	rnssi 5600	Subclass inference for range. (Contributed by Peter Mazsa, 24-Sep-2022.)
⊢ 𝐴 ⊆ 𝐵    ⇒   ⊢ ran 𝐴 ⊆ ran 𝐵