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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | fvsn 5301 | The value of a singleton of an ordered pair is the second member. (Contributed by NM, 12-Aug-1994.) |
⊢ A ∈ V & ⊢ B ∈ V ⇒ ⊢ ({〈A, B〉}‘A) = B | ||
Theorem | fvsng 5302 | The value of a singleton of an ordered pair is the second member. (Contributed by NM, 26-Oct-2012.) |
⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ({〈A, B〉}‘A) = B) | ||
Theorem | fvsnun1 5303 | The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5304. (Contributed by NM, 23-Sep-2007.) |
⊢ A ∈ V & ⊢ B ∈ V & ⊢ 𝐺 = ({〈A, B〉} ∪ (𝐹 ↾ (𝐶 ∖ {A}))) ⇒ ⊢ (𝐺‘A) = B | ||
Theorem | fvsnun2 5304 | The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5303. (Contributed by NM, 23-Sep-2007.) |
⊢ A ∈ V & ⊢ B ∈ V & ⊢ 𝐺 = ({〈A, B〉} ∪ (𝐹 ↾ (𝐶 ∖ {A}))) ⇒ ⊢ (𝐷 ∈ (𝐶 ∖ {A}) → (𝐺‘𝐷) = (𝐹‘𝐷)) | ||
Theorem | fsnunf 5305 | Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {〈𝑋, 𝑌〉}):(𝑆 ∪ {𝑋})⟶𝑇) | ||
Theorem | fsnunfv 5306 | Recover the added point from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) (Revised by NM, 18-May-2017.) |
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ ¬ 𝑋 ∈ dom 𝐹) → ((𝐹 ∪ {〈𝑋, 𝑌〉})‘𝑋) = 𝑌) | ||
Theorem | fsnunres 5307 | Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
⊢ ((𝐹 Fn 𝑆 ∧ ¬ 𝑋 ∈ 𝑆) → ((𝐹 ∪ {〈𝑋, 𝑌〉}) ↾ 𝑆) = 𝐹) | ||
Theorem | fvpr1 5308 | The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) |
⊢ A ∈ V & ⊢ 𝐶 ∈ V ⇒ ⊢ (A ≠ B → ({〈A, 𝐶〉, 〈B, 𝐷〉}‘A) = 𝐶) | ||
Theorem | fvpr2 5309 | The value of a function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) |
⊢ B ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ (A ≠ B → ({〈A, 𝐶〉, 〈B, 𝐷〉}‘B) = 𝐷) | ||
Theorem | fvpr1g 5310 | The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
⊢ ((A ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ∧ A ≠ B) → ({〈A, 𝐶〉, 〈B, 𝐷〉}‘A) = 𝐶) | ||
Theorem | fvpr2g 5311 | The value of a function with a domain of (at most) two elements. (Contributed by Alexander van der Vekens, 3-Dec-2017.) |
⊢ ((B ∈ 𝑉 ∧ 𝐷 ∈ 𝑊 ∧ A ≠ B) → ({〈A, 𝐶〉, 〈B, 𝐷〉}‘B) = 𝐷) | ||
Theorem | fvtp1g 5312 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
⊢ (((A ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (A ≠ B ∧ A ≠ 𝐶)) → ({〈A, 𝐷〉, 〈B, 𝐸〉, 〈𝐶, 𝐹〉}‘A) = 𝐷) | ||
Theorem | fvtp2g 5313 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
⊢ (((B ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (A ≠ B ∧ B ≠ 𝐶)) → ({〈A, 𝐷〉, 〈B, 𝐸〉, 〈𝐶, 𝐹〉}‘B) = 𝐸) | ||
Theorem | fvtp3g 5314 | The value of a function with a domain of (at most) three elements. (Contributed by Alexander van der Vekens, 4-Dec-2017.) |
⊢ (((𝐶 ∈ 𝑉 ∧ 𝐹 ∈ 𝑊) ∧ (A ≠ 𝐶 ∧ B ≠ 𝐶)) → ({〈A, 𝐷〉, 〈B, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐶) = 𝐹) | ||
Theorem | fvtp1 5315 | The first value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
⊢ A ∈ V & ⊢ 𝐷 ∈ V ⇒ ⊢ ((A ≠ B ∧ A ≠ 𝐶) → ({〈A, 𝐷〉, 〈B, 𝐸〉, 〈𝐶, 𝐹〉}‘A) = 𝐷) | ||
Theorem | fvtp2 5316 | The second value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
⊢ B ∈ V & ⊢ 𝐸 ∈ V ⇒ ⊢ ((A ≠ B ∧ B ≠ 𝐶) → ({〈A, 𝐷〉, 〈B, 𝐸〉, 〈𝐶, 𝐹〉}‘B) = 𝐸) | ||
Theorem | fvtp3 5317 | The third value of a function with a domain of three elements. (Contributed by NM, 14-Sep-2011.) |
⊢ 𝐶 ∈ V & ⊢ 𝐹 ∈ V ⇒ ⊢ ((A ≠ 𝐶 ∧ B ≠ 𝐶) → ({〈A, 𝐷〉, 〈B, 𝐸〉, 〈𝐶, 𝐹〉}‘𝐶) = 𝐹) | ||
Theorem | fvconst2g 5318 | The value of a constant function. (Contributed by NM, 20-Aug-2005.) |
⊢ ((B ∈ 𝐷 ∧ 𝐶 ∈ A) → ((A × {B})‘𝐶) = B) | ||
Theorem | fconst2g 5319 | A constant function expressed as a cross product. (Contributed by NM, 27-Nov-2007.) |
⊢ (B ∈ 𝐶 → (𝐹:A⟶{B} ↔ 𝐹 = (A × {B}))) | ||
Theorem | fvconst2 5320 | The value of a constant function. (Contributed by NM, 16-Apr-2005.) |
⊢ B ∈ V ⇒ ⊢ (𝐶 ∈ A → ((A × {B})‘𝐶) = B) | ||
Theorem | fconst2 5321 | A constant function expressed as a cross product. (Contributed by NM, 20-Aug-1999.) |
⊢ B ∈ V ⇒ ⊢ (𝐹:A⟶{B} ↔ 𝐹 = (A × {B})) | ||
Theorem | fconstfvm 5322* | A constant function expressed in terms of its functionality, domain, and value. See also fconst2 5321. (Contributed by Jim Kingdon, 8-Jan-2019.) |
⊢ (∃y y ∈ A → (𝐹:A⟶{B} ↔ (𝐹 Fn A ∧ ∀x ∈ A (𝐹‘x) = B))) | ||
Theorem | fconst3m 5323* | Two ways to express a constant function. (Contributed by Jim Kingdon, 8-Jan-2019.) |
⊢ (∃x x ∈ A → (𝐹:A⟶{B} ↔ (𝐹 Fn A ∧ A ⊆ (◡𝐹 “ {B})))) | ||
Theorem | fconst4m 5324* | Two ways to express a constant function. (Contributed by NM, 8-Mar-2007.) |
⊢ (∃x x ∈ A → (𝐹:A⟶{B} ↔ (𝐹 Fn A ∧ (◡𝐹 “ {B}) = A))) | ||
Theorem | resfunexg 5325 | The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. (Contributed by NM, 7-Apr-1995.) (Revised by Mario Carneiro, 22-Jun-2013.) |
⊢ ((Fun A ∧ B ∈ 𝐶) → (A ↾ B) ∈ V) | ||
Theorem | fnex 5326 | If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 5325. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
⊢ ((𝐹 Fn A ∧ A ∈ B) → 𝐹 ∈ V) | ||
Theorem | funex 5327 | If the domain of a function exists, so the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 5326. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.) (Contributed by NM, 11-Nov-1995.) |
⊢ ((Fun 𝐹 ∧ dom 𝐹 ∈ B) → 𝐹 ∈ V) | ||
Theorem | opabex 5328* | Existence of a function expressed as class of ordered pairs. (Contributed by NM, 21-Jul-1996.) |
⊢ A ∈ V & ⊢ (x ∈ A → ∃*yφ) ⇒ ⊢ {〈x, y〉 ∣ (x ∈ A ∧ φ)} ∈ V | ||
Theorem | mptexg 5329* | If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) |
⊢ (A ∈ 𝑉 → (x ∈ A ↦ B) ∈ V) | ||
Theorem | mptex 5330* | If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by NM, 22-Apr-2005.) (Revised by Mario Carneiro, 20-Dec-2013.) |
⊢ A ∈ V ⇒ ⊢ (x ∈ A ↦ B) ∈ V | ||
Theorem | fex 5331 | If the domain of a mapping is a set, the function is a set. (Contributed by NM, 3-Oct-1999.) |
⊢ ((𝐹:A⟶B ∧ A ∈ 𝐶) → 𝐹 ∈ V) | ||
Theorem | eufnfv 5332* | A function is uniquely determined by its values. (Contributed by NM, 31-Aug-2011.) |
⊢ A ∈ V & ⊢ B ∈ V ⇒ ⊢ ∃!f(f Fn A ∧ ∀x ∈ A (f‘x) = B) | ||
Theorem | funfvima 5333 | A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.) |
⊢ ((Fun 𝐹 ∧ B ∈ dom 𝐹) → (B ∈ A → (𝐹‘B) ∈ (𝐹 “ A))) | ||
Theorem | funfvima2 5334 | A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.) |
⊢ ((Fun 𝐹 ∧ A ⊆ dom 𝐹) → (B ∈ A → (𝐹‘B) ∈ (𝐹 “ A))) | ||
Theorem | funfvima3 5335 | A class including a function contains the function's value in the image of the singleton of the argument. (Contributed by NM, 23-Mar-2004.) |
⊢ ((Fun 𝐹 ∧ 𝐹 ⊆ 𝐺) → (A ∈ dom 𝐹 → (𝐹‘A) ∈ (𝐺 “ {A}))) | ||
Theorem | fnfvima 5336 | The function value of an operand in a set is contained in the image of that set, using the Fn abbreviation. (Contributed by Stefan O'Rear, 10-Mar-2015.) |
⊢ ((𝐹 Fn A ∧ 𝑆 ⊆ A ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆)) | ||
Theorem | rexima 5337* | Existential quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
⊢ (x = (𝐹‘y) → (φ ↔ ψ)) ⇒ ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (∃x ∈ (𝐹 “ B)φ ↔ ∃y ∈ B ψ)) | ||
Theorem | ralima 5338* | Universal quantification under an image in terms of the base set. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
⊢ (x = (𝐹‘y) → (φ ↔ ψ)) ⇒ ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (∀x ∈ (𝐹 “ B)φ ↔ ∀y ∈ B ψ)) | ||
Theorem | idref 5339* |
TODO: This is the same as issref 4650 (which has a much longer proof).
Should we replace issref 4650 with this one? - NM 9-May-2016.
Two ways to state a relation is reflexive. (Adapted from Tarski.) (Contributed by FL, 15-Jan-2012.) (Proof shortened by Mario Carneiro, 3-Nov-2015.) (Proof modification is discouraged.) |
⊢ (( I ↾ A) ⊆ 𝑅 ↔ ∀x ∈ A x𝑅x) | ||
Theorem | elabrex 5340* | Elementhood in an image set. (Contributed by Mario Carneiro, 14-Jan-2014.) |
⊢ B ∈ V ⇒ ⊢ (x ∈ A → B ∈ {y ∣ ∃x ∈ A y = B}) | ||
Theorem | abrexco 5341* | Composition of two image maps 𝐶(y) and B(w). (Contributed by NM, 27-May-2013.) |
⊢ B ∈ V & ⊢ (y = B → 𝐶 = 𝐷) ⇒ ⊢ {x ∣ ∃y ∈ {z ∣ ∃w ∈ A z = B}x = 𝐶} = {x ∣ ∃w ∈ A x = 𝐷} | ||
Theorem | imaiun 5342* | The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.) |
⊢ (A “ ∪ x ∈ B 𝐶) = ∪ x ∈ B (A “ 𝐶) | ||
Theorem | imauni 5343* | The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.) |
⊢ (A “ ∪ B) = ∪ x ∈ B (A “ x) | ||
Theorem | fniunfv 5344* | The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.) |
⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ ran 𝐹) | ||
Theorem | funiunfvdm 5345* | The indexed union of a function's values is the union of its image under the index class. This theorem is a slight variation of fniunfv 5344. (Contributed by Jim Kingdon, 10-Jan-2019.) |
⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ (𝐹 “ A)) | ||
Theorem | funiunfvdmf 5346* | The indexed union of a function's values is the union of its image under the index class. This version of funiunfvdm 5345 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by Jim Kingdon, 10-Jan-2019.) |
⊢ Ⅎx𝐹 ⇒ ⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ (𝐹 “ A)) | ||
Theorem | eluniimadm 5347* | Membership in the union of an image of a function. (Contributed by Jim Kingdon, 10-Jan-2019.) |
⊢ (𝐹 Fn A → (B ∈ ∪ (𝐹 “ A) ↔ ∃x ∈ A B ∈ (𝐹‘x))) | ||
Theorem | elunirn 5348* | Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.) |
⊢ (Fun 𝐹 → (A ∈ ∪ ran 𝐹 ↔ ∃x ∈ dom 𝐹 A ∈ (𝐹‘x))) | ||
Theorem | fnunirn 5349* | Membership in a union of some function-defined family of sets. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ (𝐹 Fn 𝐼 → (A ∈ ∪ ran 𝐹 ↔ ∃x ∈ 𝐼 A ∈ (𝐹‘x))) | ||
Theorem | dff13 5350* | A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.) |
⊢ (𝐹:A–1-1→B ↔ (𝐹:A⟶B ∧ ∀x ∈ A ∀y ∈ A ((𝐹‘x) = (𝐹‘y) → x = y))) | ||
Theorem | f1veqaeq 5351 | If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.) |
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ∈ A ∧ 𝐷 ∈ A)) → ((𝐹‘𝐶) = (𝐹‘𝐷) → 𝐶 = 𝐷)) | ||
Theorem | dff13f 5352* | A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.) |
⊢ Ⅎx𝐹 & ⊢ Ⅎy𝐹 ⇒ ⊢ (𝐹:A–1-1→B ↔ (𝐹:A⟶B ∧ ∀x ∈ A ∀y ∈ A ((𝐹‘x) = (𝐹‘y) → x = y))) | ||
Theorem | f1mpt 5353* | Express injection for a mapping operation. (Contributed by Mario Carneiro, 2-Jan-2017.) |
⊢ 𝐹 = (x ∈ A ↦ 𝐶) & ⊢ (x = y → 𝐶 = 𝐷) ⇒ ⊢ (𝐹:A–1-1→B ↔ (∀x ∈ A 𝐶 ∈ B ∧ ∀x ∈ A ∀y ∈ A (𝐶 = 𝐷 → x = y))) | ||
Theorem | f1fveq 5354 | Equality of function values for a one-to-one function. (Contributed by NM, 11-Feb-1997.) |
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ∈ A ∧ 𝐷 ∈ A)) → ((𝐹‘𝐶) = (𝐹‘𝐷) ↔ 𝐶 = 𝐷)) | ||
Theorem | f1elima 5355 | Membership in the image of a 1-1 map. (Contributed by Jeff Madsen, 2-Sep-2009.) |
⊢ ((𝐹:A–1-1→B ∧ 𝑋 ∈ A ∧ 𝑌 ⊆ A) → ((𝐹‘𝑋) ∈ (𝐹 “ 𝑌) ↔ 𝑋 ∈ 𝑌)) | ||
Theorem | f1imass 5356 | Taking images under a one-to-one function preserves subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ⊆ A ∧ 𝐷 ⊆ A)) → ((𝐹 “ 𝐶) ⊆ (𝐹 “ 𝐷) ↔ 𝐶 ⊆ 𝐷)) | ||
Theorem | f1imaeq 5357 | Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ⊆ A ∧ 𝐷 ⊆ A)) → ((𝐹 “ 𝐶) = (𝐹 “ 𝐷) ↔ 𝐶 = 𝐷)) | ||
Theorem | f1imapss 5358 | Taking images under a one-to-one function preserves proper subsets. (Contributed by Stefan O'Rear, 30-Oct-2014.) |
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ⊆ A ∧ 𝐷 ⊆ A)) → ((𝐹 “ 𝐶) ⊊ (𝐹 “ 𝐷) ↔ 𝐶 ⊊ 𝐷)) | ||
Theorem | dff1o6 5359* | A one-to-one onto function in terms of function values. (Contributed by NM, 29-Mar-2008.) |
⊢ (𝐹:A–1-1-onto→B ↔ (𝐹 Fn A ∧ ran 𝐹 = B ∧ ∀x ∈ A ∀y ∈ A ((𝐹‘x) = (𝐹‘y) → x = y))) | ||
Theorem | f1ocnvfv1 5360 | The converse value of the value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
⊢ ((𝐹:A–1-1-onto→B ∧ 𝐶 ∈ A) → (◡𝐹‘(𝐹‘𝐶)) = 𝐶) | ||
Theorem | f1ocnvfv2 5361 | The value of the converse value of a one-to-one onto function. (Contributed by NM, 20-May-2004.) |
⊢ ((𝐹:A–1-1-onto→B ∧ 𝐶 ∈ B) → (𝐹‘(◡𝐹‘𝐶)) = 𝐶) | ||
Theorem | f1ocnvfv 5362 | Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by Raph Levien, 10-Apr-2004.) |
⊢ ((𝐹:A–1-1-onto→B ∧ 𝐶 ∈ A) → ((𝐹‘𝐶) = 𝐷 → (◡𝐹‘𝐷) = 𝐶)) | ||
Theorem | f1ocnvfvb 5363 | Relationship between the value of a one-to-one onto function and the value of its converse. (Contributed by NM, 20-May-2004.) |
⊢ ((𝐹:A–1-1-onto→B ∧ 𝐶 ∈ A ∧ 𝐷 ∈ B) → ((𝐹‘𝐶) = 𝐷 ↔ (◡𝐹‘𝐷) = 𝐶)) | ||
Theorem | f1ocnvdm 5364 | The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006.) |
⊢ ((𝐹:A–1-1-onto→B ∧ 𝐶 ∈ B) → (◡𝐹‘𝐶) ∈ A) | ||
Theorem | f1ocnvfvrneq 5365 | If the values of a one-to-one function for two arguments from the range of the function are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.) |
⊢ ((𝐹:A–1-1→B ∧ (𝐶 ∈ ran 𝐹 ∧ 𝐷 ∈ ran 𝐹)) → ((◡𝐹‘𝐶) = (◡𝐹‘𝐷) → 𝐶 = 𝐷)) | ||
Theorem | fcof1 5366 | An application is injective if a retraction exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 11-Nov-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
⊢ ((𝐹:A⟶B ∧ (𝑅 ∘ 𝐹) = ( I ↾ A)) → 𝐹:A–1-1→B) | ||
Theorem | fcofo 5367 | An application is surjective if a section exists. Proposition 8 of [BourbakiEns] p. E.II.18. (Contributed by FL, 17-Nov-2011.) (Proof shortened by Mario Carneiro, 27-Dec-2014.) |
⊢ ((𝐹:A⟶B ∧ 𝑆:B⟶A ∧ (𝐹 ∘ 𝑆) = ( I ↾ B)) → 𝐹:A–onto→B) | ||
Theorem | cbvfo 5368* | Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
⊢ ((𝐹‘x) = y → (φ ↔ ψ)) ⇒ ⊢ (𝐹:A–onto→B → (∀x ∈ A φ ↔ ∀y ∈ B ψ)) | ||
Theorem | cbvexfo 5369* | Change bound variable between domain and range of function. (Contributed by NM, 23-Feb-1997.) |
⊢ ((𝐹‘x) = y → (φ ↔ ψ)) ⇒ ⊢ (𝐹:A–onto→B → (∃x ∈ A φ ↔ ∃y ∈ B ψ)) | ||
Theorem | cocan1 5370 | An injection is left-cancelable. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 21-Mar-2015.) |
⊢ ((𝐹:B–1-1→𝐶 ∧ 𝐻:A⟶B ∧ 𝐾:A⟶B) → ((𝐹 ∘ 𝐻) = (𝐹 ∘ 𝐾) ↔ 𝐻 = 𝐾)) | ||
Theorem | cocan2 5371 | A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.) |
⊢ ((𝐹:A–onto→B ∧ 𝐻 Fn B ∧ 𝐾 Fn B) → ((𝐻 ∘ 𝐹) = (𝐾 ∘ 𝐹) ↔ 𝐻 = 𝐾)) | ||
Theorem | fcof1o 5372 | Show that two functions are inverse to each other by computing their compositions. (Contributed by Mario Carneiro, 21-Mar-2015.) |
⊢ (((𝐹:A⟶B ∧ 𝐺:B⟶A) ∧ ((𝐹 ∘ 𝐺) = ( I ↾ B) ∧ (𝐺 ∘ 𝐹) = ( I ↾ A))) → (𝐹:A–1-1-onto→B ∧ ◡𝐹 = 𝐺)) | ||
Theorem | foeqcnvco 5373 | Condition for function equality in terms of vanishing of the composition with the converse. EDITORIAL: Is there a relation-algebraic proof of this? (Contributed by Stefan O'Rear, 12-Feb-2015.) |
⊢ ((𝐹:A–onto→B ∧ 𝐺:A–onto→B) → (𝐹 = 𝐺 ↔ (𝐹 ∘ ◡𝐺) = ( I ↾ B))) | ||
Theorem | f1eqcocnv 5374 | Condition for function equality in terms of vanishing of the composition with the inverse. (Contributed by Stefan O'Rear, 12-Feb-2015.) |
⊢ ((𝐹:A–1-1→B ∧ 𝐺:A–1-1→B) → (𝐹 = 𝐺 ↔ (◡𝐹 ∘ 𝐺) = ( I ↾ A))) | ||
Theorem | fliftrel 5375* | 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ 〈A, B〉) & ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅) & ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆) ⇒ ⊢ (φ → 𝐹 ⊆ (𝑅 × 𝑆)) | ||
Theorem | fliftel 5376* | Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ 〈A, B〉) & ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅) & ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆) ⇒ ⊢ (φ → (𝐶𝐹𝐷 ↔ ∃x ∈ 𝑋 (𝐶 = A ∧ 𝐷 = B))) | ||
Theorem | fliftel1 5377* | Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ 〈A, B〉) & ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅) & ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆) ⇒ ⊢ ((φ ∧ x ∈ 𝑋) → A𝐹B) | ||
Theorem | fliftcnv 5378* | Converse of the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ 〈A, B〉) & ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅) & ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆) ⇒ ⊢ (φ → ◡𝐹 = ran (x ∈ 𝑋 ↦ 〈B, A〉)) | ||
Theorem | fliftfun 5379* | The function 𝐹 is the unique function defined by 𝐹‘A = B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ 〈A, B〉) & ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅) & ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆) & ⊢ (x = y → A = 𝐶) & ⊢ (x = y → B = 𝐷) ⇒ ⊢ (φ → (Fun 𝐹 ↔ ∀x ∈ 𝑋 ∀y ∈ 𝑋 (A = 𝐶 → B = 𝐷))) | ||
Theorem | fliftfund 5380* | The function 𝐹 is the unique function defined by 𝐹‘A = B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ 〈A, B〉) & ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅) & ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆) & ⊢ (x = y → A = 𝐶) & ⊢ (x = y → B = 𝐷) & ⊢ ((φ ∧ (x ∈ 𝑋 ∧ y ∈ 𝑋 ∧ A = 𝐶)) → B = 𝐷) ⇒ ⊢ (φ → Fun 𝐹) | ||
Theorem | fliftfuns 5381* | The function 𝐹 is the unique function defined by 𝐹‘A = B, provided that the well-definedness condition holds. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ 〈A, B〉) & ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅) & ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆) ⇒ ⊢ (φ → (Fun 𝐹 ↔ ∀y ∈ 𝑋 ∀z ∈ 𝑋 (⦋y / x⦌A = ⦋z / x⦌A → ⦋y / x⦌B = ⦋z / x⦌B))) | ||
Theorem | fliftf 5382* | The domain and range of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ 〈A, B〉) & ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅) & ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆) ⇒ ⊢ (φ → (Fun 𝐹 ↔ 𝐹:ran (x ∈ 𝑋 ↦ A)⟶𝑆)) | ||
Theorem | fliftval 5383* | The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) |
⊢ 𝐹 = ran (x ∈ 𝑋 ↦ 〈A, B〉) & ⊢ ((φ ∧ x ∈ 𝑋) → A ∈ 𝑅) & ⊢ ((φ ∧ x ∈ 𝑋) → B ∈ 𝑆) & ⊢ (x = 𝑌 → A = 𝐶) & ⊢ (x = 𝑌 → B = 𝐷) & ⊢ (φ → Fun 𝐹) ⇒ ⊢ ((φ ∧ 𝑌 ∈ 𝑋) → (𝐹‘𝐶) = 𝐷) | ||
Theorem | isoeq1 5384 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
⊢ (𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐺 Isom 𝑅, 𝑆 (A, B))) | ||
Theorem | isoeq2 5385 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
⊢ (𝑅 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑇, 𝑆 (A, B))) | ||
Theorem | isoeq3 5386 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
⊢ (𝑆 = 𝑇 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, 𝑇 (A, B))) | ||
Theorem | isoeq4 5387 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
⊢ (A = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, 𝑆 (𝐶, B))) | ||
Theorem | isoeq5 5388 | Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.) |
⊢ (B = 𝐶 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, 𝑆 (A, 𝐶))) | ||
Theorem | nfiso 5389 | Bound-variable hypothesis builder for an isomorphism. (Contributed by NM, 17-May-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ Ⅎx𝐻 & ⊢ Ⅎx𝑅 & ⊢ Ⅎx𝑆 & ⊢ ℲxA & ⊢ ℲxB ⇒ ⊢ Ⅎx 𝐻 Isom 𝑅, 𝑆 (A, B) | ||
Theorem | isof1o 5390 | An isomorphism is a one-to-one onto function. (Contributed by NM, 27-Apr-2004.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → 𝐻:A–1-1-onto→B) | ||
Theorem | isorel 5391 | An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.) |
⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ (𝐶 ∈ A ∧ 𝐷 ∈ A)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷))) | ||
Theorem | isoresbr 5392* | A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.) |
⊢ ((𝐹 ↾ A) Isom 𝑅, 𝑆 (A, (𝐹 “ A)) → ∀x ∈ A ∀y ∈ A (x𝑅y → (𝐹‘x)𝑆(𝐹‘y))) | ||
Theorem | isoid 5393 | Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
⊢ ( I ↾ A) Isom 𝑅, 𝑅 (A, A) | ||
Theorem | isocnv 5394 | Converse law for isomorphism. Proposition 6.30(2) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) → ◡𝐻 Isom 𝑆, 𝑅 (B, A)) | ||
Theorem | isocnv2 5395 | Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom ◡𝑅, ◡𝑆(A, B)) | ||
Theorem | isores2 5396 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom 𝑅, (𝑆 ∩ (B × B))(A, B)) | ||
Theorem | isores1 5397 | An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.) |
⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐻 Isom (𝑅 ∩ (A × A)), 𝑆(A, B)) | ||
Theorem | isores3 5398 | Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.) |
⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐾 ⊆ A ∧ 𝑋 = (𝐻 “ 𝐾)) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋)) | ||
Theorem | isotr 5399 | Composition (transitive) law for isomorphism. Proposition 6.30(3) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) (Proof shortened by Mario Carneiro, 5-Dec-2016.) |
⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐺 Isom 𝑆, 𝑇 (B, 𝐶)) → (𝐺 ∘ 𝐻) Isom 𝑅, 𝑇 (A, 𝐶)) | ||
Theorem | isoini 5400 | Isomorphisms preserve initial segments. Proposition 6.31(2) of [TakeutiZaring] p. 33. (Contributed by NM, 20-Apr-2004.) |
⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐷 ∈ A) → (𝐻 “ (A ∩ (◡𝑅 “ {𝐷}))) = (B ∩ (◡𝑆 “ {(𝐻‘𝐷)}))) |
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