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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dfrn3 5801* | 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 5802* | Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵)) | ||
Theorem | elrng 5803* | Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)) | ||
Theorem | elrn2 5804* | Membership in a range. (Contributed by NM, 10-Jul-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) | ||
Theorem | elrn 5805* | Membership in a range. (Contributed by NM, 2-Apr-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴) | ||
Theorem | ssrelrn 5806* | 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 5807 | Alternate definition of domain. (Contributed by NM, 28-Dec-1996.) |
⊢ dom 𝐴 = ran ◡𝐴 | ||
Theorem | dfdmf 5808* | 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 5809 | 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 5810* | Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)) | ||
Theorem | eldm2g 5811* | 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 5812* | 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 5813* | 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 5814 | Subset theorem for domain. (Contributed by NM, 11-Aug-1994.) |
⊢ (𝐴 ⊆ 𝐵 → dom 𝐴 ⊆ dom 𝐵) | ||
Theorem | dmeq 5815 | Equality theorem for domain. (Contributed by NM, 11-Aug-1994.) |
⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵) | ||
Theorem | dmeqi 5816 | Equality inference for domain. (Contributed by NM, 4-Mar-2004.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ dom 𝐴 = dom 𝐵 | ||
Theorem | dmeqd 5817 | Equality deduction for domain. (Contributed by NM, 4-Mar-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → dom 𝐴 = dom 𝐵) | ||
Theorem | opeldmd 5818 | Membership of first of an ordered pair in a domain. Deduction version of opeldm 5819. (Contributed by AV, 11-Mar-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐴 ∈ dom 𝐶)) | ||
Theorem | opeldm 5819 | 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 5820 | 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 5821 | 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 5822 | 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 5823 | 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 5824 | 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 5825 | The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 dom 𝐵 | ||
Theorem | dmuni 5826* | 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 5827* | 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 5828* | 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 5829* | 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 5830* | 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 5831* | 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 5832 | 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 5833 | 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 5834 | The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.) |
⊢ dom V = V | ||
Theorem | dmep 5835 | 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 | domepOLD 5836 | Obsolete proof of dmep 5835 as of 26-Dec-2023. (Contributed by Scott Fenton, 27-Oct-2010.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ dom E = V | ||
Theorem | dm0rn0 5837 | An empty domain is equivalent to an empty range. (Contributed by NM, 21-May-1998.) |
⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅) | ||
Theorem | rn0 5838 | 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 5839 | 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 5840 | A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.) |
⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅)) | ||
Theorem | dmxp 5841 | 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 5842 | The domain of a Cartesian square. (Contributed by NM, 28-Jul-1995.) |
⊢ dom (𝐴 × 𝐴) = 𝐴 | ||
Theorem | dmxpin 5843 | The domain of the intersection of two Cartesian squares. Unlike in dmin 5823, equality holds. (Contributed by NM, 29-Jan-2008.) |
⊢ dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴 ∩ 𝐵) | ||
Theorem | xpid11 5844 | 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 5845 | 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 6097 gives another proof). (Contributed by NM, 8-Apr-2007.) |
⊢ dom ◡◡𝐴 = dom 𝐴 | ||
Theorem | rncnvcnv 5846 | 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 5847 | The first member of an ordered pair in a relation belongs to the domain of the relation (see op1stb 5387). (Contributed by NM, 28-Jul-2004.) |
⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴) | ||
Theorem | rneq 5848 | Equality theorem for range. (Contributed by NM, 29-Dec-1996.) |
⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵) | ||
Theorem | rneqi 5849 | Equality inference for range. (Contributed by NM, 4-Mar-2004.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ran 𝐴 = ran 𝐵 | ||
Theorem | rneqd 5850 | Equality deduction for range. (Contributed by NM, 4-Mar-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ran 𝐴 = ran 𝐵) | ||
Theorem | rnss 5851 | Subset theorem for range. (Contributed by NM, 22-Mar-1998.) |
⊢ (𝐴 ⊆ 𝐵 → ran 𝐴 ⊆ ran 𝐵) | ||
Theorem | rnssi 5852 | Subclass inference for range. (Contributed by Peter Mazsa, 24-Sep-2022.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ ran 𝐴 ⊆ ran 𝐵 | ||
Theorem | brelrng 5853 | 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 5854 | 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 5855 | 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 5856 | The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5712 and brv 5388. (Contributed by NM, 2-Jul-2008.) |
⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) | ||
Theorem | relelrn 5857 | The second argument of a binary relation belongs to its range. (Contributed by NM, 2-Jul-2008.) |
⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐵 ∈ ran 𝑅) | ||
Theorem | releldmb 5858* | Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.) |
⊢ (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | ||
Theorem | relelrnb 5859* | Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.) |
⊢ (Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴)) | ||
Theorem | releldmi 5860 | The first argument of a binary relation belongs to its domain. (Contributed by NM, 28-Apr-2015.) |
⊢ Rel 𝑅 ⇒ ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ dom 𝑅) | ||
Theorem | relelrni 5861 | The second argument of a binary relation belongs to its range. (Contributed by NM, 28-Apr-2015.) |
⊢ Rel 𝑅 ⇒ ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ ran 𝑅) | ||
Theorem | dfrnf 5862* | Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 ⇒ ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} | ||
Theorem | nfdm 5863 | Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥dom 𝐴 | ||
Theorem | nfrn 5864 | Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥ran 𝐴 | ||
Theorem | dmiin 5865 | Domain of an intersection. (Contributed by FL, 15-Oct-2012.) |
⊢ dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 dom 𝐵 | ||
Theorem | rnopab 5866* | The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.) |
⊢ ran {〈𝑥, 𝑦〉 ∣ 𝜑} = {𝑦 ∣ ∃𝑥𝜑} | ||
Theorem | rnmpt 5867* | The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} | ||
Theorem | elrnmpt 5868* | The range of a function in maps-to notation. (Contributed by Mario Carneiro, 20-Feb-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝐶 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) | ||
Theorem | elrnmpt1s 5869* | Elementhood in an image set. (Contributed by Mario Carneiro, 12-Sep-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) ⇒ ⊢ ((𝐷 ∈ 𝐴 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ ran 𝐹) | ||
Theorem | elrnmpt1 5870 | Elementhood in an image set. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹) | ||
Theorem | elrnmptg 5871* | Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)) | ||
Theorem | elrnmpti 5872* | Membership in the range of a function. (Contributed by NM, 30-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) | ||
Theorem | elrnmptd 5873* | The range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐶 ∈ ran 𝐹) | ||
Theorem | elrnmptdv 5874* | Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → 𝐷 = 𝐵) ⇒ ⊢ (𝜑 → 𝐷 ∈ ran 𝐹) | ||
Theorem | elrnmpt2d 5875* | Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝐶 ∈ ran 𝐹) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵) | ||
Theorem | dfiun3g 5876 | Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) | ||
Theorem | dfiin3g 5877 | Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) | ||
Theorem | dfiun3 5878 | Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐵 ∈ V ⇒ ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) | ||
Theorem | dfiin3 5879 | Alternate definition of indexed intersection when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐵 ∈ V ⇒ ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ ran (𝑥 ∈ 𝐴 ↦ 𝐵) | ||
Theorem | riinint 5880* | Express a relative indexed intersection as an intersection. (Contributed by Stefan O'Rear, 22-Feb-2015.) |
⊢ ((𝑋 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 𝑆 ⊆ 𝑋) → (𝑋 ∩ ∩ 𝑘 ∈ 𝐼 𝑆) = ∩ ({𝑋} ∪ ran (𝑘 ∈ 𝐼 ↦ 𝑆))) | ||
Theorem | relrn0 5881 | A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.) |
⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅)) | ||
Theorem | dmrnssfld 5882 | The domain and range of a class are included in its double union. (Contributed by NM, 13-May-2008.) |
⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 | ||
Theorem | dmcoss 5883 | Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ dom (𝐴 ∘ 𝐵) ⊆ dom 𝐵 | ||
Theorem | rncoss 5884 | Range of a composition. (Contributed by NM, 19-Mar-1998.) |
⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴 | ||
Theorem | dmcosseq 5885 | Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) = dom 𝐵) | ||
Theorem | dmcoeq 5886 | Domain of a composition. (Contributed by NM, 19-Mar-1998.) |
⊢ (dom 𝐴 = ran 𝐵 → dom (𝐴 ∘ 𝐵) = dom 𝐵) | ||
Theorem | rncoeq 5887 | Range of a composition. (Contributed by NM, 19-Mar-1998.) |
⊢ (dom 𝐴 = ran 𝐵 → ran (𝐴 ∘ 𝐵) = ran 𝐴) | ||
Theorem | reseq1 5888 | Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) | ||
Theorem | reseq2 5889 | Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) | ||
Theorem | reseq1i 5890 | Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶) | ||
Theorem | reseq2i 5891 | Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵) | ||
Theorem | reseq12i 5892 | Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷) | ||
Theorem | reseq1d 5893 | Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) | ||
Theorem | reseq2d 5894 | Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) | ||
Theorem | reseq12d 5895 | Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷)) | ||
Theorem | nfres 5896 | Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵) | ||
Theorem | csbres 5897 | Distribute proper substitution through the restriction of a class. (Contributed by Alan Sare, 10-Nov-2012.) (Revised by NM, 23-Aug-2018.) |
⊢ ⦋𝐴 / 𝑥⦌(𝐵 ↾ 𝐶) = (⦋𝐴 / 𝑥⦌𝐵 ↾ ⦋𝐴 / 𝑥⦌𝐶) | ||
Theorem | res0 5898 | A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.) |
⊢ (𝐴 ↾ ∅) = ∅ | ||
Theorem | dfres3 5899 | Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × ran 𝐴)) | ||
Theorem | opelres 5900 | Ordered pair elementhood in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.) (Revised by BJ, 18-Feb-2022.) Commute the consequent. (Revised by Peter Mazsa, 24-Sep-2022.) |
⊢ (𝐶 ∈ 𝑉 → (〈𝐵, 𝐶〉 ∈ (𝑅 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 〈𝐵, 𝐶〉 ∈ 𝑅))) |
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