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Type | Label | Description |
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Statement | ||
Theorem | fvtresfn 7001* | Functionality of a tuple-restriction function. (Contributed by Stefan O'Rear, 24-Jan-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) ⇒ ⊢ (𝑋 ∈ 𝐵 → (𝐹‘𝑋) = (𝑋 ↾ 𝑉)) | ||
Theorem | fvmpts 7002* | Value of a function given in maps-to notation, using explicit class substitution. (Contributed by Scott Fenton, 17-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐶 ↦ 𝐵) ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → (𝐹‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵) | ||
Theorem | fvmpt3 7003* | Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) & ⊢ (𝑥 ∈ 𝐷 → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶) | ||
Theorem | fvmpt3i 7004* | Value of a function given in maps-to notation, with a slightly different sethood condition. (Contributed by Mario Carneiro, 11-Sep-2015.) |
⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 ∈ 𝐷 → (𝐹‘𝐴) = 𝐶) | ||
Theorem | fvmptdf 7005* | Deduction version of fvmptd 7006 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by AV, 29-Mar-2024.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) | ||
Theorem | fvmptd 7006* | Deduction version of fvmpt 6999. (Contributed by Scott Fenton, 18-Feb-2013.) (Revised by Mario Carneiro, 31-Aug-2015.) (Proof shortened by AV, 29-Mar-2024.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) | ||
Theorem | fvmptd2 7007* | Deduction version of fvmpt 6999 (where the definition of the mapping does not depend on the common antecedent 𝜑). (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) | ||
Theorem | mptrcl 7008* | Reverse closure for a mapping: If the function value of a mapping has a member, the argument belongs to the base class of the mapping. (Contributed by AV, 4-Apr-2020.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝐼 ∈ (𝐹‘𝑋) → 𝑋 ∈ 𝐴) | ||
Theorem | fvmpt2i 7009* | Value of a function given by the maps-to notation. (Contributed by Mario Carneiro, 23-Apr-2014.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝑥 ∈ 𝐴 → (𝐹‘𝑥) = ( I ‘𝐵)) | ||
Theorem | fvmpt2 7010* | Value of a function given by the maps-to notation. (Contributed by FL, 21-Jun-2010.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐹‘𝑥) = 𝐵) | ||
Theorem | fvmptss 7011* | If all the values of the mapping are subsets of a class 𝐶, then so is any evaluation of the mapping, even if 𝐷 is not in the base set 𝐴. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (𝐹‘𝐷) ⊆ 𝐶) | ||
Theorem | fvmpt2d 7012* | Deduction version of fvmpt2 7010. (Contributed by Thierry Arnoux, 8-Dec-2016.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵) | ||
Theorem | fvmptex 7013* | Express a function 𝐹 whose value 𝐵 may not always be a set in terms of another function 𝐺 for which sethood is guaranteed. (Note that ( I ‘𝐵) is just shorthand for if(𝐵 ∈ V, 𝐵, ∅), and it is always a set by fvex 6905.) Note also that these functions are not the same; wherever 𝐵(𝐶) is not a set, 𝐶 is not in the domain of 𝐹 (so it evaluates to the empty set), but 𝐶 is in the domain of 𝐺, and 𝐺(𝐶) is defined to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 23-Apr-2014.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵)) ⇒ ⊢ (𝐹‘𝐶) = (𝐺‘𝐶) | ||
Theorem | fvmptd3f 7014* | Alternate deduction version of fvmpt 6999 with three nonfreeness hypotheses instead of distinct variable conditions. (Contributed by AV, 19-Jan-2022.) |
⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓)) & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜑 ⇒ ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) | ||
Theorem | fvmptd2f 7015* | Alternate deduction version of fvmpt 6999, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) (Proof shortened by AV, 19-Jan-2022.) |
⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓)) & ⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) | ||
Theorem | fvmptdv 7016* | Alternate deduction version of fvmpt 6999, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓)) ⇒ ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) | ||
Theorem | fvmptdv2 7017* | Alternate deduction version of fvmpt 6999, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) |
⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = 𝐶)) | ||
Theorem | mpteqb 7018* | Bidirectional equality theorem for a mapping abstraction. Equivalent to eqfnfv 7033. (Contributed by Mario Carneiro, 14-Nov-2014.) |
⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ((𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶)) | ||
Theorem | fvmptt 7019* | Closed theorem form of fvmpt 6999. (Contributed by Scott Fenton, 21-Feb-2013.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ ((∀𝑥(𝑥 = 𝐴 → 𝐵 = 𝐶) ∧ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ∧ (𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉)) → (𝐹‘𝐴) = 𝐶) | ||
Theorem | fvmptf 7020* | Value of a function given by an ordered-pair class abstraction. This version of fvmptg 6997 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ⇒ ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑉) → (𝐹‘𝐴) = 𝐶) | ||
Theorem | fvmptnf 7021* | The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvmptn 7023 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑥𝐶 & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) ⇒ ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐴) = ∅) | ||
Theorem | fvmptd3 7022* | Deduction version of fvmpt 6999. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) & ⊢ (𝜑 → 𝐴 ∈ 𝐷) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) | ||
Theorem | fvmptn 7023* | This somewhat non-intuitive theorem tells us the value of its function is the empty set when the class 𝐶 it would otherwise map to is a proper class. This is a technical lemma that can help eliminate redundant sethood antecedents otherwise required by fvmptg 6997. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 9-Sep-2013.) |
⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (¬ 𝐶 ∈ V → (𝐹‘𝐷) = ∅) | ||
Theorem | fvmptss2 7024* | A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.) |
⊢ (𝑥 = 𝐷 → 𝐵 = 𝐶) & ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (𝐹‘𝐷) ⊆ 𝐶 | ||
Theorem | elfvmptrab1w 7025* | Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. Version of elfvmptrab1 7026 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Alexander van der Vekens, 15-Jul-2018.) Avoid ax-13 2372. (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) & ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V) ⇒ ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)) | ||
Theorem | elfvmptrab1 7026* | Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker elfvmptrab1w 7025 when possible. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (New usage is discouraged.) |
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ ⦋𝑥 / 𝑚⦌𝑀 ∣ 𝜑}) & ⊢ (𝑋 ∈ 𝑉 → ⦋𝑋 / 𝑚⦌𝑀 ∈ V) ⇒ ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ ⦋𝑋 / 𝑚⦌𝑀)) | ||
Theorem | elfvmptrab 7027* | Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.) |
⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑀 ∣ 𝜑}) & ⊢ (𝑋 ∈ 𝑉 → 𝑀 ∈ V) ⇒ ⊢ (𝑌 ∈ (𝐹‘𝑋) → (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑀)) | ||
Theorem | fvopab4ndm 7028* | Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by NM, 28-Mar-2008.) |
⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⇒ ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐹‘𝐵) = ∅) | ||
Theorem | fvmptndm 7029* | Value of a function given by the maps-to notation, outside of its domain. (Contributed by AV, 31-Dec-2020.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ (¬ 𝑋 ∈ 𝐴 → (𝐹‘𝑋) = ∅) | ||
Theorem | fvmptrabfv 7030* | Value of a function mapping a set to a class abstraction restricting the value of another function. (Contributed by AV, 18-Feb-2022.) |
⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑}) & ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} | ||
Theorem | fvopab5 7031* | The value of a function that is expressed as an ordered pair abstraction. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑦𝜓)) | ||
Theorem | fvopab6 7032* | Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.) |
⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐵)} & ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐶 ∈ 𝑅 ∧ 𝜓) → (𝐹‘𝐴) = 𝐶) | ||
Theorem | eqfnfv 7033* | Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) | ||
Theorem | eqfnfv2 7034* | Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥)))) | ||
Theorem | eqfnfv3 7035* | Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ (𝐹‘𝑥) = (𝐺‘𝑥))))) | ||
Theorem | eqfnfvd 7036* | Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐴) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐺‘𝑥)) ⇒ ⊢ (𝜑 → 𝐹 = 𝐺) | ||
Theorem | eqfnfv2f 7037* | Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 7033 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐺 ⇒ ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = (𝐺‘𝑥))) | ||
Theorem | eqfunfv 7038* | Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.) |
⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 = 𝐺 ↔ (dom 𝐹 = dom 𝐺 ∧ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = (𝐺‘𝑥)))) | ||
Theorem | eqfnun 7039 | Two functions on 𝐴 ∪ 𝐵 are equal if and only if they have equal restrictions to both 𝐴 and 𝐵. (Contributed by Jeff Madsen, 19-Jun-2011.) |
⊢ ((𝐹 Fn (𝐴 ∪ 𝐵) ∧ 𝐺 Fn (𝐴 ∪ 𝐵)) → (𝐹 = 𝐺 ↔ ((𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴) ∧ (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)))) | ||
Theorem | fvreseq0 7040* | Equality of restricted functions is determined by their values (for functions with different domains). (Contributed by AV, 6-Jan-2019.) |
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐶) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐶)) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) | ||
Theorem | fvreseq1 7041* | Equality of a function restricted to the domain of another function. (Contributed by AV, 6-Jan-2019.) |
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵) ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ↾ 𝐵) = 𝐺 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) | ||
Theorem | fvreseq 7042* | Equality of restricted functions is determined by their values. (Contributed by NM, 3-Aug-1994.) (Proof shortened by AV, 4-Mar-2019.) |
⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) | ||
Theorem | fnmptfvd 7043* | A function with a given domain is a mapping defined by its function values. (Contributed by AV, 1-Mar-2019.) |
⊢ (𝜑 → 𝑀 Fn 𝐴) & ⊢ (𝑖 = 𝑎 → 𝐷 = 𝐶) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝐴) → 𝐷 ∈ 𝑈) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐶 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑀 = (𝑎 ∈ 𝐴 ↦ 𝐶) ↔ ∀𝑖 ∈ 𝐴 (𝑀‘𝑖) = 𝐷)) | ||
Theorem | fndmdif 7044* | Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ≠ (𝐺‘𝑥)}) | ||
Theorem | fndmdifcom 7045 | The difference set between two functions is commutative. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∖ 𝐺) = dom (𝐺 ∖ 𝐹)) | ||
Theorem | fndmdifeq0 7046 | The difference set of two functions is empty if and only if the functions are equal. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (dom (𝐹 ∖ 𝐺) = ∅ ↔ 𝐹 = 𝐺)) | ||
Theorem | fndmin 7047* | Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → dom (𝐹 ∩ 𝐺) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = (𝐺‘𝑥)}) | ||
Theorem | fneqeql 7048 | Two functions are equal iff their equalizer is the whole domain. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ dom (𝐹 ∩ 𝐺) = 𝐴)) | ||
Theorem | fneqeql2 7049 | Two functions are equal iff their equalizer contains the whole domain. (Contributed by Stefan O'Rear, 9-Mar-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ 𝐴 ⊆ dom (𝐹 ∩ 𝐺))) | ||
Theorem | fnreseql 7050 | Two functions are equal on a subset iff their equalizer contains that subset. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴 ∧ 𝑋 ⊆ 𝐴) → ((𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ↔ 𝑋 ⊆ dom (𝐹 ∩ 𝐺))) | ||
Theorem | chfnrn 7051* | The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by NM, 31-Aug-1999.) |
⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥) → ran 𝐹 ⊆ ∪ 𝐴) | ||
Theorem | funfvop 7052 | Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 14-Oct-1996.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ⟨𝐴, (𝐹‘𝐴)⟩ ∈ 𝐹) | ||
Theorem | funfvbrb 7053 | Two ways to say that 𝐴 is in the domain of 𝐹. (Contributed by Mario Carneiro, 1-May-2014.) |
⊢ (Fun 𝐹 → (𝐴 ∈ dom 𝐹 ↔ 𝐴𝐹(𝐹‘𝐴))) | ||
Theorem | fvimacnvi 7054 | A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (◡𝐹 “ 𝐵)) → (𝐹‘𝐴) ∈ 𝐵) | ||
Theorem | fvimacnv 7055 | The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 6632 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵))) | ||
Theorem | funimass3 7056 | A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 7055 would be the special case of 𝐴 being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵))) | ||
Theorem | funimass5 7057* | A subclass of a preimage in terms of function values. (Contributed by NM, 15-May-2007.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐴 ⊆ (◡𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) | ||
Theorem | funconstss 7058* | Two ways of specifying that a function is constant on a subdomain. (Contributed by NM, 8-Mar-2007.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ {𝐵}))) | ||
Theorem | fvimacnvALT 7059 | Alternate proof of fvimacnv 7055, based on funimass3 7056. If funimass3 7056 is ever proved directly, as opposed to using funimacnv 6630 pointwise, then the proof of funimacnv 6630 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) ∈ 𝐵 ↔ 𝐴 ∈ (◡𝐹 “ 𝐵))) | ||
Theorem | elpreima 7060 | Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ (◡𝐹 “ 𝐶) ↔ (𝐵 ∈ 𝐴 ∧ (𝐹‘𝐵) ∈ 𝐶))) | ||
Theorem | elpreimad 7061 | Membership in the preimage of a set under a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) & ⊢ (𝜑 → (𝐹‘𝐵) ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐵 ∈ (◡𝐹 “ 𝐶)) | ||
Theorem | fniniseg 7062 | Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.) (Proof shortened by Mario Carneiro , 28-Apr-2015.) |
⊢ (𝐹 Fn 𝐴 → (𝐶 ∈ (◡𝐹 “ {𝐵}) ↔ (𝐶 ∈ 𝐴 ∧ (𝐹‘𝐶) = 𝐵))) | ||
Theorem | fncnvima2 7063* | Inverse images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ 𝐵) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) ∈ 𝐵}) | ||
Theorem | fniniseg2 7064* | Inverse point images under functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {𝐵}) = {𝑥 ∈ 𝐴 ∣ (𝐹‘𝑥) = 𝐵}) | ||
Theorem | unpreima 7065 | Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∪ 𝐵)) = ((◡𝐹 “ 𝐴) ∪ (◡𝐹 “ 𝐵))) | ||
Theorem | inpreima 7066 | Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jun-2016.) |
⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵))) | ||
Theorem | difpreima 7067 | Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.) |
⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∖ 𝐵)) = ((◡𝐹 “ 𝐴) ∖ (◡𝐹 “ 𝐵))) | ||
Theorem | respreima 7068 | The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐵) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ 𝐵)) | ||
Theorem | cnvimainrn 7069 | The preimage of the intersection of the range of a class and a class 𝐴 is the preimage of the class 𝐴. (Contributed by AV, 17-Sep-2024.) |
⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = (◡𝐹 “ 𝐴)) | ||
Theorem | sspreima 7070 | The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ 𝐵) → (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ 𝐵)) | ||
Theorem | iinpreima 7071* | Preimage of an intersection. (Contributed by FL, 16-Apr-2012.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵)) | ||
Theorem | intpreima 7072* | Preimage of an intersection. (Contributed by FL, 28-Apr-2012.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ≠ ∅) → (◡𝐹 “ ∩ 𝐴) = ∩ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝑥)) | ||
Theorem | fimacnvOLD 7073 | Obsolete version of fimacnv 6740 as of 20-Sep-2024. (Contributed by FL, 25-Jan-2007.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴) | ||
Theorem | fimacnvinrn 7074 | Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 21-Jan-2017.) |
⊢ (Fun 𝐹 → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ ran 𝐹))) | ||
Theorem | fimacnvinrn2 7075 | Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 17-Feb-2017.) |
⊢ ((Fun 𝐹 ∧ ran 𝐹 ⊆ 𝐵) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (𝐴 ∩ 𝐵))) | ||
Theorem | rescnvimafod 7076 | The restriction of a function to a preimage of a class is a function onto the intersection of this class and the range of the function. (Contributed by AV, 13-Sep-2024.) (Revised by AV, 29-Sep-2024.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐸 = (ran 𝐹 ∩ 𝐵)) & ⊢ (𝜑 → 𝐷 = (◡𝐹 “ 𝐵)) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝐷):𝐷–onto→𝐸) | ||
Theorem | fvn0ssdmfun 7077* | If a class' function values for certain arguments is not the empty set, the arguments are contained in the domain of the class, and the class restricted to the arguments is a function, analogous to fvfundmfvn0 6935. (Contributed by AV, 27-Jan-2020.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ (∀𝑎 ∈ 𝐷 (𝐹‘𝑎) ≠ ∅ → (𝐷 ⊆ dom 𝐹 ∧ Fun (𝐹 ↾ 𝐷))) | ||
Theorem | fnopfv 7078 | Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by NM, 30-Sep-2004.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ⟨𝐵, (𝐹‘𝐵)⟩ ∈ 𝐹) | ||
Theorem | fvelrn 7079 | A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹) | ||
Theorem | nelrnfvne 7080 | A function value cannot be any element not contained in the range of the function. (Contributed by AV, 28-Jan-2020.) |
⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹 ∧ 𝑌 ∉ ran 𝐹) → (𝐹‘𝑋) ≠ 𝑌) | ||
Theorem | fveqdmss 7081* | If the empty set is not contained in the range of a function, and the function values of another class (not necessarily a function) are equal to the function values of the function for all elements of the domain of the function, then the domain of the function is contained in the domain of the class. (Contributed by AV, 28-Jan-2020.) |
⊢ 𝐷 = dom 𝐵 ⇒ ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → 𝐷 ⊆ dom 𝐴) | ||
Theorem | fveqressseq 7082* | If the empty set is not contained in the range of a function, and the function values of another class (not necessarily a function) are equal to the function values of the function for all elements of the domain of the function, then the class restricted to the domain of the function is the function itself. (Contributed by AV, 28-Jan-2020.) |
⊢ 𝐷 = dom 𝐵 ⇒ ⊢ ((Fun 𝐵 ∧ ∅ ∉ ran 𝐵 ∧ ∀𝑥 ∈ 𝐷 (𝐴‘𝑥) = (𝐵‘𝑥)) → (𝐴 ↾ 𝐷) = 𝐵) | ||
Theorem | fnfvelrn 7083 | A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.) |
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐹‘𝐵) ∈ ran 𝐹) | ||
Theorem | ffvelcdm 7084 | A function's value belongs to its codomain. (Contributed by NM, 12-Aug-1999.) |
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐵) | ||
Theorem | fnfvelrnd 7085 | A function's value belongs to its range. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐵 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐹‘𝐵) ∈ ran 𝐹) | ||
Theorem | ffvelcdmi 7086 | A function's value belongs to its codomain. (Contributed by NM, 6-Apr-2005.) |
⊢ 𝐹:𝐴⟶𝐵 ⇒ ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) ∈ 𝐵) | ||
Theorem | ffvelcdmda 7087 | A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐹‘𝐶) ∈ 𝐵) | ||
Theorem | ffvelcdmd 7088 | A function's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016.) |
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) & ⊢ (𝜑 → 𝐶 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐹‘𝐶) ∈ 𝐵) | ||
Theorem | rexrn 7089* | Restricted existential quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.) |
⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐹 Fn 𝐴 → (∃𝑥 ∈ ran 𝐹𝜑 ↔ ∃𝑦 ∈ 𝐴 𝜓)) | ||
Theorem | ralrn 7090* | Restricted universal quantification over the range of a function. (Contributed by Mario Carneiro, 24-Dec-2013.) (Revised by Mario Carneiro, 20-Aug-2014.) |
⊢ (𝑥 = (𝐹‘𝑦) → (𝜑 ↔ 𝜓)) ⇒ ⊢ (𝐹 Fn 𝐴 → (∀𝑥 ∈ ran 𝐹𝜑 ↔ ∀𝑦 ∈ 𝐴 𝜓)) | ||
Theorem | elrnrexdm 7091* | For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 8-Dec-2017.) |
⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥))) | ||
Theorem | elrnrexdmb 7092* | For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 17-Dec-2017.) |
⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥))) | ||
Theorem | eldmrexrn 7093* | For any element in the domain of a function there is an element in the range of the function which is the function value for the element of the domain. (Contributed by Alexander van der Vekens, 8-Dec-2017.) |
⊢ (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌))) | ||
Theorem | eldmrexrnb 7094* | For any element in the domain of a function, there is an element in the range of the function which is the value of the function at that element. Because of the definition df-fv 6552 of the value of a function, the theorem is only valid in general if the empty set is not contained in the range of the function (the implication "to the right" is always valid). Indeed, with the definition df-fv 6552 of the value of a function, (𝐹‘𝑌) = ∅ may mean that the value of 𝐹 at 𝑌 is the empty set or that 𝐹 is not defined at 𝑌. (Contributed by Alexander van der Vekens, 17-Dec-2017.) |
⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 ↔ ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌))) | ||
Theorem | fvcofneq 7095* | The values of two function compositions are equal if the values of the composed functions are pairwise equal. (Contributed by AV, 26-Jan-2019.) |
⊢ ((𝐺 Fn 𝐴 ∧ 𝐾 Fn 𝐵) → ((𝑋 ∈ (𝐴 ∩ 𝐵) ∧ (𝐺‘𝑋) = (𝐾‘𝑋) ∧ ∀𝑥 ∈ (ran 𝐺 ∩ ran 𝐾)(𝐹‘𝑥) = (𝐻‘𝑥)) → ((𝐹 ∘ 𝐺)‘𝑋) = ((𝐻 ∘ 𝐾)‘𝑋))) | ||
Theorem | ralrnmptw 7096* | A restricted quantifier over an image set. Version of ralrnmpt 7098 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2372. (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | rexrnmptw 7097* | A restricted quantifier over an image set. Version of rexrnmpt 7099 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 20-Aug-2015.) Avoid ax-13 2372. (Revised by Gino Giotto, 26-Jan-2024.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | ralrnmpt 7098* | A restricted quantifier over an image set. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker ralrnmptw 7096 when possible. (Contributed by Mario Carneiro, 20-Aug-2015.) (New usage is discouraged.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∀𝑦 ∈ ran 𝐹𝜓 ↔ ∀𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | rexrnmpt 7099* | A restricted quantifier over an image set. Usage of this theorem is discouraged because it depends on ax-13 2372. Use the weaker rexrnmptw 7097 when possible. (Contributed by Mario Carneiro, 20-Aug-2015.) (New usage is discouraged.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) & ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (∃𝑦 ∈ ran 𝐹𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜒)) | ||
Theorem | f0cli 7100 | Unconditional closure of a function when the codomain includes the empty set. (Contributed by Mario Carneiro, 12-Sep-2013.) |
⊢ 𝐹:𝐴⟶𝐵 & ⊢ ∅ ∈ 𝐵 ⇒ ⊢ (𝐹‘𝐶) ∈ 𝐵 |
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