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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | djussxp 5801* | 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 5802* | Version of ralxp 5797 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 5803* | Version of rexxp 5798 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 5804* | 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 5805* | Deduce equality of a relation and an ordered-pair class abstraction. Compare abbi2dv 2871. (Contributed by NM, 24-Feb-2014.) |
⊢ Rel 𝐴 & ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 𝜓)) ⇒ ⊢ (𝜑 → 𝐴 = {〈𝑥, 𝑦〉 ∣ 𝜓}) | ||
Theorem | relop 5806* | A necessary and sufficient condition for a Kuratowski ordered pair to be a relation. (Contributed by NM, 3-Jun-2008.) A relation is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a relation is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is relsnopg 5759, as funsng 6552 is to funop 7095. (New usage is discouraged.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (Rel 〈𝐴, 𝐵〉 ↔ ∃𝑥∃𝑦(𝐴 = {𝑥} ∧ 𝐵 = {𝑥, 𝑦})) | ||
Theorem | ideqg 5807 | 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 5808 | 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 5809 | 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 5810 | 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 5811 | Subclass theorem for composition. (Contributed by FL, 30-Dec-2010.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∘ 𝐶) ⊆ (𝐵 ∘ 𝐶)) | ||
Theorem | coss2 5812 | Subclass theorem for composition. (Contributed by NM, 5-Apr-2013.) |
⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∘ 𝐴) ⊆ (𝐶 ∘ 𝐵)) | ||
Theorem | coeq1 5813 | Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.) |
⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)) | ||
Theorem | coeq2 5814 | Equality theorem for composition of two classes. (Contributed by NM, 3-Jan-1997.) |
⊢ (𝐴 = 𝐵 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)) | ||
Theorem | coeq1i 5815 | Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶) | ||
Theorem | coeq2i 5816 | Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵) | ||
Theorem | coeq1d 5817 | Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)) | ||
Theorem | coeq2d 5818 | Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)) | ||
Theorem | coeq12i 5819 | Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷) | ||
Theorem | coeq12d 5820 | 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 5821 | Bound-variable hypothesis builder for function value. (Contributed by NM, 1-Sep-1999.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ∘ 𝐵) | ||
Theorem | brcog 5822* | 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 5823* | 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 5824 | 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 5825* | 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 5826* | 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 5827* | 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 5828 | Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Kyle Wyonch, 27-Apr-2021.) |
⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) | ||
Theorem | cnveq 5829 | Equality theorem for converse relation. (Contributed by NM, 13-Aug-1995.) |
⊢ (𝐴 = 𝐵 → ◡𝐴 = ◡𝐵) | ||
Theorem | cnveqi 5830 | Equality inference for converse relation. (Contributed by NM, 23-Dec-2008.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ◡𝐴 = ◡𝐵 | ||
Theorem | cnveqd 5831 | Equality deduction for converse relation. (Contributed by NM, 6-Dec-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ◡𝐴 = ◡𝐵) | ||
Theorem | elcnv 5832* | 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 5833* | 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 5834 | 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 5835 | 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 5836 | 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 5837 | Ordered-pair membership in converse relation. (Contributed by NM, 13-Aug-1995.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ ◡𝑅 ↔ 〈𝐵, 𝐴〉 ∈ 𝑅) | ||
Theorem | brcnv 5838 | 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 5839 | Move class substitution in and out of the converse of a relation. Version of csbcnvgALT 5840 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 5840 | Move class substitution in and out of the converse of a relation. Version of csbcnv 5839 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 5841 | 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 5842* | 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 5843* | 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 5844* | Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.) |
⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} | ||
Theorem | dfrn3 5845* | 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 5846* | Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵)) | ||
Theorem | elrng 5847* | Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)) | ||
Theorem | elrn2 5848* | Membership in a range. (Contributed by NM, 10-Jul-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) | ||
Theorem | elrn 5849* | Membership in a range. (Contributed by NM, 2-Apr-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴) | ||
Theorem | ssrelrn 5850* | 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 5851 | Alternate definition of domain. (Contributed by NM, 28-Dec-1996.) |
⊢ dom 𝐴 = ran ◡𝐴 | ||
Theorem | dfdmf 5852* | 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 5853 | 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 5854* | Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)) | ||
Theorem | eldm2g 5855* | 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 5856* | 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 5857* | 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 5858 | Subset theorem for domain. (Contributed by NM, 11-Aug-1994.) |
⊢ (𝐴 ⊆ 𝐵 → dom 𝐴 ⊆ dom 𝐵) | ||
Theorem | dmeq 5859 | Equality theorem for domain. (Contributed by NM, 11-Aug-1994.) |
⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵) | ||
Theorem | dmeqi 5860 | Equality inference for domain. (Contributed by NM, 4-Mar-2004.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ dom 𝐴 = dom 𝐵 | ||
Theorem | dmeqd 5861 | Equality deduction for domain. (Contributed by NM, 4-Mar-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → dom 𝐴 = dom 𝐵) | ||
Theorem | opeldmd 5862 | Membership of first of an ordered pair in a domain. Deduction version of opeldm 5863. (Contributed by AV, 11-Mar-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐴 ∈ dom 𝐶)) | ||
Theorem | opeldm 5863 | 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 5864 | 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 5865 | 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 5866 | 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 5867 | The domain of an intersection is included in the intersection of the domains. Theorem 6 of [Suppes] p. 60. (Contributed by NM, 15-Sep-2004.) |
⊢ dom (𝐴 ∩ 𝐵) ⊆ (dom 𝐴 ∩ dom 𝐵) | ||
Theorem | breldmd 5868 | Membership of first of a binary relation in a domain. (Contributed by Glauco Siliprandi, 23-Apr-2023.) |
⊢ (𝜑 → 𝐴 ∈ 𝐶) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ dom 𝑅) | ||
Theorem | dmiun 5869 | The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 dom 𝐵 | ||
Theorem | dmuni 5870* | 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 5871* | 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 | dmopabelb 5872* | A set is an element of the domain of a ordered pair class abstraction iff there is a second set so that both sets fulfil the wff of the class abstraction. (Contributed by AV, 19-Oct-2023.) |
⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ dom {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑦𝜓)) | ||
Theorem | dmopab2rex 5873* | The domain of an ordered pair class abstraction with two nested restricted existential quantifiers. (Contributed by AV, 23-Oct-2023.) |
⊢ (∀𝑢 ∈ 𝑈 (∀𝑣 ∈ 𝑉 𝐵 ∈ 𝑋 ∧ ∀𝑖 ∈ 𝐼 𝐷 ∈ 𝑊) → dom {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑈 (∃𝑣 ∈ 𝑉 (𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∨ ∃𝑖 ∈ 𝐼 (𝑥 = 𝐶 ∧ 𝑦 = 𝐷))} = {𝑥 ∣ ∃𝑢 ∈ 𝑈 (∃𝑣 ∈ 𝑉 𝑥 = 𝐴 ∨ ∃𝑖 ∈ 𝐼 𝑥 = 𝐶)}) | ||
Theorem | dmopabss 5874* | 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 5875* | 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 | dm0 5876 | 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 5877 | 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 5878 | The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.) |
⊢ dom V = V | ||
Theorem | dmep 5879 | The domain of the membership relation is the universal class. (Contributed by Scott Fenton, 27-Oct-2010.) (Proof shortened by BJ, 26-Dec-2023.) |
⊢ dom E = V | ||
Theorem | dm0rn0 5880 | An empty domain is equivalent to an empty range. (Contributed by NM, 21-May-1998.) |
⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅) | ||
Theorem | rn0 5881 | The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) |
⊢ ran ∅ = ∅ | ||
Theorem | rnep 5882 | The range of the membership relation is the universal class minus the empty set. (Contributed by BJ, 26-Dec-2023.) |
⊢ ran E = (V ∖ {∅}) | ||
Theorem | reldm0 5883 | A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.) |
⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅)) | ||
Theorem | dmxp 5884 | 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 5885 | The domain of a Cartesian square. (Contributed by NM, 28-Jul-1995.) |
⊢ dom (𝐴 × 𝐴) = 𝐴 | ||
Theorem | dmxpin 5886 | The domain of the intersection of two Cartesian squares. Unlike in dmin 5867, equality holds. (Contributed by NM, 29-Jan-2008.) |
⊢ dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴 ∩ 𝐵) | ||
Theorem | xpid11 5887 | The Cartesian square is a one-to-one construction. (Contributed by NM, 5-Nov-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ((𝐴 × 𝐴) = (𝐵 × 𝐵) ↔ 𝐴 = 𝐵) | ||
Theorem | dmcnvcnv 5888 | The domain of the double converse of a class is equal to its domain (even when that class in not a relation, in which case dfrel2 6141 gives another proof). (Contributed by NM, 8-Apr-2007.) |
⊢ dom ◡◡𝐴 = dom 𝐴 | ||
Theorem | rncnvcnv 5889 | The range of the double converse of a class is equal to its range (even when that class in not a relation). (Contributed by NM, 8-Apr-2007.) |
⊢ ran ◡◡𝐴 = ran 𝐴 | ||
Theorem | elreldm 5890 | The first member of an ordered pair in a relation belongs to the domain of the relation (see op1stb 5428). (Contributed by NM, 28-Jul-2004.) |
⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴) | ||
Theorem | rneq 5891 | Equality theorem for range. (Contributed by NM, 29-Dec-1996.) |
⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵) | ||
Theorem | rneqi 5892 | Equality inference for range. (Contributed by NM, 4-Mar-2004.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ran 𝐴 = ran 𝐵 | ||
Theorem | rneqd 5893 | Equality deduction for range. (Contributed by NM, 4-Mar-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ran 𝐴 = ran 𝐵) | ||
Theorem | rnss 5894 | Subset theorem for range. (Contributed by NM, 22-Mar-1998.) |
⊢ (𝐴 ⊆ 𝐵 → ran 𝐴 ⊆ ran 𝐵) | ||
Theorem | rnssi 5895 | Subclass inference for range. (Contributed by Peter Mazsa, 24-Sep-2022.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ ran 𝐴 ⊆ ran 𝐵 | ||
Theorem | brelrng 5896 | The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.) |
⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶) | ||
Theorem | brelrn 5897 | The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶) | ||
Theorem | opelrn 5898 | Membership of second member of an ordered pair in a range. (Contributed by NM, 23-Feb-1997.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐵 ∈ ran 𝐶) | ||
Theorem | releldm 5899 | The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5756 and brv 5429. (Contributed by NM, 2-Jul-2008.) |
⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) | ||
Theorem | relelrn 5900 | The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.) |
⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅) |
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