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Type | Label | Description |
---|---|---|
Statement | ||
Definition | df-isom 6501* | Define the isomorphism predicate. We read this as "𝐻 is an 𝑅, 𝑆 isomorphism of 𝐴 onto 𝐵". Normally, 𝑅 and 𝑆 are ordering relations on 𝐴 and 𝐵 respectively. Definition 6.28 of [TakeutiZaring] p. 32, whose notation is the same as ours except that 𝑅 and 𝑆 are subscripts. (Contributed by NM, 4-Mar-1997.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) | ||
Theorem | dffun2 6502* | Alternate definition of a function. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2138, ax-12 2172. (Revised by SN, 19-Dec-2024.) Avoid ax-11 2155. (Revised by BTernaryTau, 29-Dec-2024.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))) | ||
Theorem | dffun2OLD 6503* | Obsolete version of dffun2 6502 as of 29-Dec-2024. (Contributed by NM, 29-Dec-1996.) Avoid ax-10 2138, ax-12 2172. (Revised by SN, 19-Dec-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))) | ||
Theorem | dffun2OLDOLD 6504* | Obsolete version of dffun2 6502 as of 11-Dec-2024. (Contributed by NM, 29-Dec-1996.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → 𝑦 = 𝑧))) | ||
Theorem | dffun6 6505* | Alternate definition of a function using "at most one" notation. (Contributed by NM, 9-Mar-1995.) Avoid ax-10 2138, ax-12 2172. (Revised by SN, 19-Dec-2024.) |
⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) | ||
Theorem | dffun3 6506* | Alternate definition of function. (Contributed by NM, 29-Dec-1996.) (Proof shortened by SN, 19-Dec-2024.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) | ||
Theorem | dffun3OLD 6507* | Obsolete version of dffun3 6506 as of 19-Dec-2024. Alternate definition of function. (Contributed by NM, 29-Dec-1996.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(𝑥𝐴𝑦 → 𝑦 = 𝑧))) | ||
Theorem | dffun4 6508* | Alternate definition of a function. Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 29-Dec-1996.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ ⟨𝑥, 𝑧⟩ ∈ 𝐴) → 𝑦 = 𝑧))) | ||
Theorem | dffun5 6509* | Alternate definition of function. (Contributed by NM, 29-Dec-1996.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑦 = 𝑧))) | ||
Theorem | dffun6f 6510* | Definition of function, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 9-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.) |
⊢ Ⅎ𝑥𝐴 & ⊢ Ⅎ𝑦𝐴 ⇒ ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦)) | ||
Theorem | dffun6OLD 6511* | Obsolete version of dffun6 6505 as of 19-Dec-2024. (Contributed by NM, 9-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦)) | ||
Theorem | funmo 6512* | A function has at most one value for each argument. (Contributed by NM, 24-May-1998.) (Proof shortened by SN, 19-Dec-2024.) |
⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦) | ||
Theorem | funmoOLD 6513* | Obsolete version of funmo 6512 as of 19-Dec-2024. (Contributed by NM, 24-May-1998.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (Fun 𝐹 → ∃*𝑦 𝐴𝐹𝑦) | ||
Theorem | funrel 6514 | A function is a relation. (Contributed by NM, 1-Aug-1994.) |
⊢ (Fun 𝐴 → Rel 𝐴) | ||
Theorem | 0nelfun 6515 | A function does not contain the empty set. (Contributed by BJ, 26-Nov-2021.) |
⊢ (Fun 𝑅 → ∅ ∉ 𝑅) | ||
Theorem | funss 6516 | Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.) |
⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴)) | ||
Theorem | funeq 6517 | Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵)) | ||
Theorem | funeqi 6518 | Equality inference for the function predicate. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (Fun 𝐴 ↔ Fun 𝐵) | ||
Theorem | funeqd 6519 | Equality deduction for the function predicate. (Contributed by NM, 23-Feb-2013.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (Fun 𝐴 ↔ Fun 𝐵)) | ||
Theorem | nffun 6520 | Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.) |
⊢ Ⅎ𝑥𝐹 ⇒ ⊢ Ⅎ𝑥Fun 𝐹 | ||
Theorem | sbcfung 6521 | Distribute proper substitution through the function predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.) |
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]Fun 𝐹 ↔ Fun ⦋𝐴 / 𝑥⦌𝐹)) | ||
Theorem | funeu 6522* | There is exactly one value of a function. (Contributed by NM, 22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
⊢ ((Fun 𝐹 ∧ 𝐴𝐹𝐵) → ∃!𝑦 𝐴𝐹𝑦) | ||
Theorem | funeu2 6523* | There is exactly one value of a function. (Contributed by NM, 3-Aug-1994.) |
⊢ ((Fun 𝐹 ∧ ⟨𝐴, 𝐵⟩ ∈ 𝐹) → ∃!𝑦⟨𝐴, 𝑦⟩ ∈ 𝐹) | ||
Theorem | dffun7 6524* | Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. (Enderton's definition is ambiguous because "there is only one" could mean either "there is at most one" or "there is exactly one". However, dffun8 6525 shows that it does not matter which meaning we pick.) (Contributed by NM, 4-Nov-2002.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 𝑥𝐴𝑦)) | ||
Theorem | dffun8 6525* | Alternate definition of a function. One possibility for the definition of a function in [Enderton] p. 42. Compare dffun7 6524. (Contributed by NM, 4-Nov-2002.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃!𝑦 𝑥𝐴𝑦)) | ||
Theorem | dffun9 6526* | Alternate definition of a function. (Contributed by NM, 28-Mar-2007.) (Revised by NM, 16-Jun-2017.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥 ∈ dom 𝐴∃*𝑦 ∈ ran 𝐴 𝑥𝐴𝑦)) | ||
Theorem | funfn 6527 | A class is a function if and only if it is a function on its domain. (Contributed by NM, 13-Aug-2004.) |
⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴) | ||
Theorem | funfnd 6528 | A function is a function on its domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
⊢ (𝜑 → Fun 𝐴) ⇒ ⊢ (𝜑 → 𝐴 Fn dom 𝐴) | ||
Theorem | funi 6529 | The identity relation is a function. Part of Theorem 10.4 of [Quine] p. 65. See also idfn 6625. (Contributed by NM, 30-Apr-1998.) |
⊢ Fun I | ||
Theorem | nfunv 6530 | The universal class is not a function. (Contributed by Raph Levien, 27-Jan-2004.) |
⊢ ¬ Fun V | ||
Theorem | funopg 6531 | A Kuratowski ordered pair of sets is a function only if its components are equal. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 26-Apr-2015.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 6548, as relsnopg 5756 is to relop 5803. (New usage is discouraged.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ Fun ⟨𝐴, 𝐵⟩) → 𝐴 = 𝐵) | ||
Theorem | funopab 6532* | A class of ordered pairs is a function when there is at most one second member for each pair. (Contributed by NM, 16-May-1995.) |
⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∀𝑥∃*𝑦𝜑) | ||
Theorem | funopabeq 6533* | A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.) |
⊢ Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} | ||
Theorem | funopab4 6534* | A class of ordered pairs of values in the form used by df-mpt 5188 is a function. (Contributed by NM, 17-Feb-2013.) |
⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)} | ||
Theorem | funmpt 6535 | A function in maps-to notation is a function. (Contributed by Mario Carneiro, 13-Jan-2013.) |
⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | ||
Theorem | funmpt2 6536 | Functionality of a class given by a maps-to notation. (Contributed by FL, 17-Feb-2008.) (Revised by Mario Carneiro, 31-May-2014.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) ⇒ ⊢ Fun 𝐹 | ||
Theorem | funco 6537 | The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (Contributed by NM, 26-Jan-1997.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
⊢ ((Fun 𝐹 ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | ||
Theorem | funresfunco 6538 | Composition of two functions, generalization of funco 6537. (Contributed by Alexander van der Vekens, 25-Jul-2017.) |
⊢ ((Fun (𝐹 ↾ ran 𝐺) ∧ Fun 𝐺) → Fun (𝐹 ∘ 𝐺)) | ||
Theorem | funres 6539 | A restriction of a function is a function. Compare Exercise 18 of [TakeutiZaring] p. 25. (Contributed by NM, 16-Aug-1994.) |
⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴)) | ||
Theorem | funresd 6540 | A restriction of a function is a function. (Contributed by Glauco Siliprandi, 2-Jan-2022.) |
⊢ (𝜑 → Fun 𝐹) ⇒ ⊢ (𝜑 → Fun (𝐹 ↾ 𝐴)) | ||
Theorem | funssres 6541 | The restriction of a function to the domain of a subclass equals the subclass. (Contributed by NM, 15-Aug-1994.) |
⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺) | ||
Theorem | fun2ssres 6542 | Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.) |
⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴)) | ||
Theorem | funun 6543 | The union of functions with disjoint domains is a function. Theorem 4.6 of [Monk1] p. 43. (Contributed by NM, 12-Aug-1994.) |
⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺)) | ||
Theorem | fununmo 6544* | If the union of classes is a function, there is at most one element in relation to an arbitrary element regarding one of these classes. (Contributed by AV, 18-Jul-2019.) |
⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐹𝑦) | ||
Theorem | fununfun 6545 | If the union of classes is a function, the classes itselves are functions. (Contributed by AV, 18-Jul-2019.) |
⊢ (Fun (𝐹 ∪ 𝐺) → (Fun 𝐹 ∧ Fun 𝐺)) | ||
Theorem | fundif 6546 | A function with removed elements is still a function. (Contributed by AV, 7-Jun-2021.) |
⊢ (Fun 𝐹 → Fun (𝐹 ∖ 𝐴)) | ||
Theorem | funcnvsn 6547 | The converse singleton of an ordered pair is a function. This is equivalent to funsn 6550 via cnvsn 6175, but stating it this way allows to skip the sethood assumptions on 𝐴 and 𝐵. (Contributed by NM, 30-Apr-2015.) |
⊢ Fun ◡{⟨𝐴, 𝐵⟩} | ||
Theorem | funsng 6548 | A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 28-Jun-2011.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {⟨𝐴, 𝐵⟩}) | ||
Theorem | fnsng 6549 | Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.) |
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩} Fn {𝐴}) | ||
Theorem | funsn 6550 | A singleton of an ordered pair is a function. Theorem 10.5 of [Quine] p. 65. (Contributed by NM, 12-Aug-1994.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ Fun {⟨𝐴, 𝐵⟩} | ||
Theorem | funprg 6551 | A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}) | ||
Theorem | funtpg 6552 | A set of three pairs is a function if their first members are different. (Contributed by Alexander van der Vekens, 5-Dec-2017.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → Fun {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩}) | ||
Theorem | funpr 6553 | A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (𝐴 ≠ 𝐵 → Fun {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}) | ||
Theorem | funtp 6554 | A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V & ⊢ 𝐸 ∈ V & ⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → Fun {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}) | ||
Theorem | fnsn 6555 | Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ {⟨𝐴, 𝐵⟩} Fn {𝐴} | ||
Theorem | fnprg 6556 | Function with a domain of two different values. (Contributed by FL, 26-Jun-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) ∧ 𝐴 ≠ 𝐵) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} Fn {𝐴, 𝐵}) | ||
Theorem | fntpg 6557 | Function with a domain of three different values. (Contributed by Alexander van der Vekens, 5-Dec-2017.) |
⊢ (((𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) ∧ (𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐶 ∈ 𝐻) ∧ (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍)) → {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩, ⟨𝑍, 𝐶⟩} Fn {𝑋, 𝑌, 𝑍}) | ||
Theorem | fntp 6558 | A function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) (Revised by Mario Carneiro, 26-Apr-2015.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐶 ∈ V & ⊢ 𝐷 ∈ V & ⊢ 𝐸 ∈ V & ⊢ 𝐹 ∈ V ⇒ ⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} Fn {𝐴, 𝐵, 𝐶}) | ||
Theorem | funcnvpr 6559 | The converse pair of ordered pairs is a function if the second members are different. Note that the second members need not be sets. (Contributed by AV, 23-Jan-2021.) |
⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}) | ||
Theorem | funcnvtp 6560 | The converse triple of ordered pairs is a function if the second members are pairwise different. Note that the second members need not be sets. (Contributed by AV, 23-Jan-2021.) |
⊢ (((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐷 ≠ 𝐹)) → Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩}) | ||
Theorem | funcnvqp 6561 | The converse quadruple of ordered pairs is a function if the second members are pairwise different. Note that the second members need not be sets. (Contributed by AV, 23-Jan-2021.) (Proof shortened by JJ, 14-Jul-2021.) |
⊢ ((((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉) ∧ (𝐸 ∈ 𝑊 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝐵 ≠ 𝐷 ∧ 𝐵 ≠ 𝐹 ∧ 𝐵 ≠ 𝐻) ∧ (𝐷 ≠ 𝐹 ∧ 𝐷 ≠ 𝐻) ∧ 𝐹 ≠ 𝐻)) → Fun ◡({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩})) | ||
Theorem | fun0 6562 | The empty set is a function. Theorem 10.3 of [Quine] p. 65. (Contributed by NM, 7-Apr-1998.) |
⊢ Fun ∅ | ||
Theorem | funcnv0 6563 | The converse of the empty set is a function. (Contributed by AV, 7-Jan-2021.) |
⊢ Fun ◡∅ | ||
Theorem | funcnvcnv 6564 | The double converse of a function is a function. (Contributed by NM, 21-Sep-2004.) |
⊢ (Fun 𝐴 → Fun ◡◡𝐴) | ||
Theorem | funcnv2 6565* | A simpler equivalence for single-rooted (see funcnv 6566). (Contributed by NM, 9-Aug-2004.) |
⊢ (Fun ◡𝐴 ↔ ∀𝑦∃*𝑥 𝑥𝐴𝑦) | ||
Theorem | funcnv 6566* | The converse of a class is a function iff the class is single-rooted, which means that for any 𝑦 in the range of 𝐴 there is at most one 𝑥 such that 𝑥𝐴𝑦. Definition of single-rooted in [Enderton] p. 43. See funcnv2 6565 for a simpler version. (Contributed by NM, 13-Aug-2004.) |
⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃*𝑥 𝑥𝐴𝑦) | ||
Theorem | funcnv3 6567* | A condition showing a class is single-rooted. (See funcnv 6566). (Contributed by NM, 26-May-2006.) |
⊢ (Fun ◡𝐴 ↔ ∀𝑦 ∈ ran 𝐴∃!𝑥 ∈ dom 𝐴 𝑥𝐴𝑦) | ||
Theorem | fun2cnv 6568* | The double converse of a class is a function iff the class is single-valued. Each side is equivalent to Definition 6.4(2) of [TakeutiZaring] p. 23, who use the notation "Un(A)" for single-valued. Note that 𝐴 is not necessarily a function. (Contributed by NM, 13-Aug-2004.) |
⊢ (Fun ◡◡𝐴 ↔ ∀𝑥∃*𝑦 𝑥𝐴𝑦) | ||
Theorem | svrelfun 6569 | A single-valued relation is a function. (See fun2cnv 6568 for "single-valued.") Definition 6.4(4) of [TakeutiZaring] p. 24. (Contributed by NM, 17-Jan-2006.) |
⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ Fun ◡◡𝐴)) | ||
Theorem | fncnv 6570* | Single-rootedness (see funcnv 6566) of a class cut down by a Cartesian product. (Contributed by NM, 5-Mar-2007.) |
⊢ (◡(𝑅 ∩ (𝐴 × 𝐵)) Fn 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃!𝑥 ∈ 𝐴 𝑥𝑅𝑦) | ||
Theorem | fun11 6571* | Two ways of stating that 𝐴 is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one (but not necessarily a function). (Contributed by NM, 17-Jan-2006.) |
⊢ ((Fun ◡◡𝐴 ∧ Fun ◡𝐴) ↔ ∀𝑥∀𝑦∀𝑧∀𝑤((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑤) → (𝑥 = 𝑧 ↔ 𝑦 = 𝑤))) | ||
Theorem | fununi 6572* | The union of a chain (with respect to inclusion) of functions is a function. (Contributed by NM, 10-Aug-2004.) |
⊢ (∀𝑓 ∈ 𝐴 (Fun 𝑓 ∧ ∀𝑔 ∈ 𝐴 (𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓)) → Fun ∪ 𝐴) | ||
Theorem | funin 6573 | The intersection with a function is a function. Exercise 14(a) of [Enderton] p. 53. (Contributed by NM, 19-Mar-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
⊢ (Fun 𝐹 → Fun (𝐹 ∩ 𝐺)) | ||
Theorem | funres11 6574 | The restriction of a one-to-one function is one-to-one. (Contributed by NM, 25-Mar-1998.) |
⊢ (Fun ◡𝐹 → Fun ◡(𝐹 ↾ 𝐴)) | ||
Theorem | funcnvres 6575 | The converse of a restricted function. (Contributed by NM, 27-Mar-1998.) |
⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ (𝐹 “ 𝐴))) | ||
Theorem | cnvresid 6576 | Converse of a restricted identity function. (Contributed by FL, 4-Mar-2007.) |
⊢ ◡( I ↾ 𝐴) = ( I ↾ 𝐴) | ||
Theorem | funcnvres2 6577 | The converse of a restriction of the converse of a function equals the function restricted to the image of its converse. (Contributed by NM, 4-May-2005.) |
⊢ (Fun 𝐹 → ◡(◡𝐹 ↾ 𝐴) = (𝐹 ↾ (◡𝐹 “ 𝐴))) | ||
Theorem | funimacnv 6578 | The image of the preimage of a function. (Contributed by NM, 25-May-2004.) |
⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐴)) = (𝐴 ∩ ran 𝐹)) | ||
Theorem | funimass1 6579 | A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ ran 𝐹) → ((◡𝐹 “ 𝐴) ⊆ 𝐵 → 𝐴 ⊆ (𝐹 “ 𝐵))) | ||
Theorem | funimass2 6580 | A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.) |
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ (◡𝐹 “ 𝐵)) → (𝐹 “ 𝐴) ⊆ 𝐵) | ||
Theorem | imadif 6581 | The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.) |
⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∖ 𝐵)) = ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵))) | ||
Theorem | imain 6582 | The image of an intersection is the intersection of images. (Contributed by Paul Chapman, 11-Apr-2009.) |
⊢ (Fun ◡𝐹 → (𝐹 “ (𝐴 ∩ 𝐵)) = ((𝐹 “ 𝐴) ∩ (𝐹 “ 𝐵))) | ||
Theorem | funimaexg 6583 | Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 10-Sep-2006.) Shorten proof and avoid ax-10 2138, ax-12 2172. (Revised by SN, 19-Dec-2024.) |
⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V) | ||
Theorem | funimaexgOLD 6584 | Obsolete version of funimaexg 6583 as of 19-Dec-2024. (Contributed by NM, 10-Sep-2006.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V) | ||
Theorem | funimaex 6585 | The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5241. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 17-Nov-2002.) |
⊢ 𝐵 ∈ V ⇒ ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V) | ||
Theorem | isarep1 6586* | Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by 𝜑(𝑥, 𝑦) i.e. the class ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴). If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.) (Proof shortened by SN, 19-Dec-2024.) |
⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑) | ||
Theorem | isarep1OLD 6587* | Obsolete version of isarep1 6586 as of 19-Dec-2024. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝑏 ∈ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 [𝑏 / 𝑦]𝜑) | ||
Theorem | isarep2 6588* | Part of a study of the Axiom of Replacement used by the Isabelle prover. In Isabelle, the sethood of PrimReplace is apparently postulated implicitly by its type signature "[ i, [ i, i ] => o ] => i", which automatically asserts that it is a set without using any axioms. To prove that it is a set in Metamath, we need the hypotheses of Isabelle's "Axiom of Replacement" as well as the Axiom of Replacement in the form funimaex 6585. (Contributed by NM, 26-Oct-2006.) |
⊢ 𝐴 ∈ V & ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦∀𝑧((𝜑 ∧ [𝑧 / 𝑦]𝜑) → 𝑦 = 𝑧) ⇒ ⊢ ∃𝑤 𝑤 = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} “ 𝐴) | ||
Theorem | fneq1 6589 | Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) | ||
Theorem | fneq2 6590 | Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵)) | ||
Theorem | fneq1d 6591 | Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝜑 → 𝐹 = 𝐺) ⇒ ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) | ||
Theorem | fneq2d 6592 | Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵)) | ||
Theorem | fneq12d 6593 | Equality deduction for function predicate with domain. (Contributed by NM, 26-Jun-2011.) |
⊢ (𝜑 → 𝐹 = 𝐺) & ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐵)) | ||
Theorem | fneq12 6594 | Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.) |
⊢ ((𝐹 = 𝐺 ∧ 𝐴 = 𝐵) → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐵)) | ||
Theorem | fneq1i 6595 | Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.) |
⊢ 𝐹 = 𝐺 ⇒ ⊢ (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴) | ||
Theorem | fneq2i 6596 | Equality inference for function predicate with domain. (Contributed by NM, 4-Sep-2011.) |
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵) | ||
Theorem | nffn 6597 | Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.) |
⊢ Ⅎ𝑥𝐹 & ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 | ||
Theorem | fnfun 6598 | A function with domain is a function. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | ||
Theorem | fnfund 6599 | A function with domain is a function, deduction form. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
⊢ (𝜑 → 𝐹 Fn 𝐴) ⇒ ⊢ (𝜑 → Fun 𝐹) | ||
Theorem | fnrel 6600 | A function with domain is a relation. (Contributed by NM, 1-Aug-1994.) |
⊢ (𝐹 Fn 𝐴 → Rel 𝐹) |
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