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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dfrn2 5901* | Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.) |
⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} | ||
Theorem | dfrn3 5902* | 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 5903* | Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵)) | ||
Theorem | elrng 5904* | Membership in a range. (Contributed by Scott Fenton, 2-Feb-2011.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)) | ||
Theorem | elrn2 5905* | Membership in a range. (Contributed by NM, 10-Jul-1994.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥〈𝑥, 𝐴〉 ∈ 𝐵) | ||
Theorem | elrn 5906* | Membership in a range. (Contributed by NM, 2-Apr-2004.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴) | ||
Theorem | ssrelrn 5907* | 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 5908 | Alternate definition of domain. (Contributed by NM, 28-Dec-1996.) |
⊢ dom 𝐴 = ran ◡𝐴 | ||
Theorem | dfdmf 5909* | 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 5910 | 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 5911* | Domain membership. Theorem 4 of [Suppes] p. 59. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom 𝐵 ↔ ∃𝑦 𝐴𝐵𝑦)) | ||
Theorem | eldm2g 5912* | 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 5913* | 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 5914* | 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 5915 | Subset theorem for domain. (Contributed by NM, 11-Aug-1994.) |
⊢ (𝐴 ⊆ 𝐵 → dom 𝐴 ⊆ dom 𝐵) | ||
Theorem | dmeq 5916 | Equality theorem for domain. (Contributed by NM, 11-Aug-1994.) |
⊢ (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵) | ||
Theorem | dmeqi 5917 | Equality inference for domain. (Contributed by NM, 4-Mar-2004.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ dom 𝐴 = dom 𝐵 | ||
Theorem | dmeqd 5918 | Equality deduction for domain. (Contributed by NM, 4-Mar-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → dom 𝐴 = dom 𝐵) | ||
Theorem | opeldmd 5919 | Membership of first of an ordered pair in a domain. Deduction version of opeldm 5920. (Contributed by AV, 11-Mar-2021.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → (〈𝐴, 𝐵〉 ∈ 𝐶 → 𝐴 ∈ dom 𝐶)) | ||
Theorem | opeldm 5920 | 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 5921 | 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 5922 | 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 5923 | 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 5924 | 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 5925 | 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 5926 | The domain of an indexed union. (Contributed by Mario Carneiro, 26-Apr-2016.) |
⊢ dom ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 dom 𝐵 | ||
Theorem | dmuni 5927* | 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 5928* | 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 5929* | 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 5930* | 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 5931* | 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 5932* | 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 5933 | 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 5934 | 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 5935 | The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.) |
⊢ dom V = V | ||
Theorem | dmep 5936 | 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 5937 | An empty domain is equivalent to an empty range. (Contributed by NM, 21-May-1998.) |
⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅) | ||
Theorem | rn0 5938 | 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 5939 | 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 5940 | A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.) |
⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅)) | ||
Theorem | dmxp 5941 | 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.) Avoid ax-10 2138, ax-11 2154, ax-12 2174. (Revised by SN, 12-Aug-2025.) |
⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) | ||
Theorem | dmxpOLD 5942 | Obsolete version of dmxp 5941 as of 19-Dec-2024. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) | ||
Theorem | dmxpid 5943 | The domain of a Cartesian square. (Contributed by NM, 28-Jul-1995.) |
⊢ dom (𝐴 × 𝐴) = 𝐴 | ||
Theorem | dmxpin 5944 | The domain of the intersection of two Cartesian squares. Unlike in dmin 5924, equality holds. (Contributed by NM, 29-Jan-2008.) |
⊢ dom ((𝐴 × 𝐴) ∩ (𝐵 × 𝐵)) = (𝐴 ∩ 𝐵) | ||
Theorem | xpid11 5945 | 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 5946 | 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 6210 gives another proof). (Contributed by NM, 8-Apr-2007.) |
⊢ dom ◡◡𝐴 = dom 𝐴 | ||
Theorem | rncnvcnv 5947 | 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 5948 | The first member of an ordered pair in a relation belongs to the domain of the relation (see op1stb 5481). (Contributed by NM, 28-Jul-2004.) |
⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → ∩ ∩ 𝐵 ∈ dom 𝐴) | ||
Theorem | rneq 5949 | Equality theorem for range. (Contributed by NM, 29-Dec-1996.) |
⊢ (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵) | ||
Theorem | rneqi 5950 | Equality inference for range. (Contributed by NM, 4-Mar-2004.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ ran 𝐴 = ran 𝐵 | ||
Theorem | rneqd 5951 | Equality deduction for range. (Contributed by NM, 4-Mar-2004.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ran 𝐴 = ran 𝐵) | ||
Theorem | rnss 5952 | Subset theorem for range. (Contributed by NM, 22-Mar-1998.) |
⊢ (𝐴 ⊆ 𝐵 → ran 𝐴 ⊆ ran 𝐵) | ||
Theorem | rnssi 5953 | Subclass inference for range. (Contributed by Peter Mazsa, 24-Sep-2022.) |
⊢ 𝐴 ⊆ 𝐵 ⇒ ⊢ ran 𝐴 ⊆ ran 𝐵 | ||
Theorem | brelrng 5954 | 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 5955 | 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 5956 | 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 5957 | The first argument of a binary relation belongs to its domain. Note that 𝐴𝑅𝐵 does not imply Rel 𝑅: see for example nrelv 5812 and brv 5482. (Contributed by NM, 2-Jul-2008.) |
⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅) | ||
Theorem | relelrn 5958 | 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 5959* | Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.) |
⊢ (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥)) | ||
Theorem | relelrnb 5960* | Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.) |
⊢ (Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴)) | ||
Theorem | releldmi 5961 | 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 5962 | 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 5963* | 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 5964 | Bound-variable hypothesis builder for domain. (Contributed by NM, 30-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥dom 𝐴 | ||
Theorem | nfrn 5965 | Bound-variable hypothesis builder for range. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥ran 𝐴 | ||
Theorem | dmiin 5966 | Domain of an intersection. (Contributed by FL, 15-Oct-2012.) |
⊢ dom ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ∩ 𝑥 ∈ 𝐴 dom 𝐵 | ||
Theorem | rnopab 5967* | 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 | rnopabss 5968* | Upper bound for the range of a restricted class of ordered pairs. (Contributed by Eric Schmidt, 16-Sep-2025.) |
⊢ ran {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 | ||
Theorem | rnopab3 5969* | The range of a restricted class of ordered pairs. (Contributed by Eric Schmidt, 16-Sep-2025.) |
⊢ (∀𝑦 ∈ 𝐴 ∃𝑥𝜑 ↔ ran {〈𝑥, 𝑦〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = 𝐴) | ||
Theorem | rnmpt 5970* | 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 5971* | 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 5972* | 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 5973 | 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 5974* | 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 5975* | 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 5976* | 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 | elrnmpt1d 5977 | Elementhood in an image set. Deducion version of elrnmpt1 5973. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝜑 → 𝑥 ∈ 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐵 ∈ ran 𝐹) | ||
Theorem | elrnmptdv 5978* | 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 5979* | 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 5980 | 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 5981 | 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 5982 | 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 5983 | 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 5984* | 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 5985 | A relation is empty iff its range is empty. (Contributed by NM, 15-Sep-2004.) |
⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ran 𝐴 = ∅)) | ||
Theorem | dmrnssfld 5986 | 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 5987 | 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 5988 | Range of a composition. (Contributed by NM, 19-Mar-1998.) |
⊢ ran (𝐴 ∘ 𝐵) ⊆ ran 𝐴 | ||
Theorem | dmcosseq 5989 | Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) Avoid ax-11 2154. (Revised by BTernaryTau, 23-Jun-2025.) |
⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) = dom 𝐵) | ||
Theorem | dmcosseqOLD 5990 | Obsolete version of dmcosseq 5989 as of 23-Jun-2025. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴 ∘ 𝐵) = dom 𝐵) | ||
Theorem | dmcoeq 5991 | Domain of a composition. (Contributed by NM, 19-Mar-1998.) |
⊢ (dom 𝐴 = ran 𝐵 → dom (𝐴 ∘ 𝐵) = dom 𝐵) | ||
Theorem | rncoeq 5992 | Range of a composition. (Contributed by NM, 19-Mar-1998.) |
⊢ (dom 𝐴 = ran 𝐵 → ran (𝐴 ∘ 𝐵) = ran 𝐴) | ||
Theorem | reseq1 5993 | Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) | ||
Theorem | reseq2 5994 | Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) | ||
Theorem | reseq1i 5995 | Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶) | ||
Theorem | reseq2i 5996 | Equality inference for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵) | ||
Theorem | reseq12i 5997 | Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.) |
⊢ 𝐴 = 𝐵 & ⊢ 𝐶 = 𝐷 ⇒ ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷) | ||
Theorem | reseq1d 5998 | Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶)) | ||
Theorem | reseq2d 5999 | Equality deduction for restrictions. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐶 ↾ 𝐴) = (𝐶 ↾ 𝐵)) | ||
Theorem | reseq12d 6000 | Equality deduction for restrictions. (Contributed by NM, 21-Oct-2014.) |
⊢ (𝜑 → 𝐴 = 𝐵) & ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷)) |
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