Theorem List for Intuitionistic Logic Explorer - 5301-5400 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | fresin 5301 |
An identity for the mapping relationship under restriction. (Contributed
by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro,
26-May-2016.)
|
⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ↾ 𝑋):(𝐴 ∩ 𝑋)⟶𝐵) |
|
Theorem | resasplitss 5302 |
If two functions agree on their common domain, their union contains a
union of three functions with pairwise disjoint domains. If we assumed
the law of the excluded middle, this would be equality rather than subset.
(Contributed by Jim Kingdon, 28-Dec-2018.)
|
⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐹 ↾ (𝐴 ∩ 𝐵)) = (𝐺 ↾ (𝐴 ∩ 𝐵))) → ((𝐹 ↾ (𝐴 ∩ 𝐵)) ∪ ((𝐹 ↾ (𝐴 ∖ 𝐵)) ∪ (𝐺 ↾ (𝐵 ∖ 𝐴)))) ⊆ (𝐹 ∪ 𝐺)) |
|
Theorem | fcoi1 5303 |
Composition of a mapping and restricted identity. (Contributed by NM,
13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|
⊢ (𝐹:𝐴⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐴)) = 𝐹) |
|
Theorem | fcoi2 5304 |
Composition of restricted identity and a mapping. (Contributed by NM,
13-Dec-2003.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|
⊢ (𝐹:𝐴⟶𝐵 → (( I ↾ 𝐵) ∘ 𝐹) = 𝐹) |
|
Theorem | feu 5305* |
There is exactly one value of a function in its codomain. (Contributed
by NM, 10-Dec-2003.)
|
⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐶 ∈ 𝐴) → ∃!𝑦 ∈ 𝐵 〈𝐶, 𝑦〉 ∈ 𝐹) |
|
Theorem | fcnvres 5306 |
The converse of a restriction of a function. (Contributed by NM,
26-Mar-1998.)
|
⊢ (𝐹:𝐴⟶𝐵 → ◡(𝐹 ↾ 𝐴) = (◡𝐹 ↾ 𝐵)) |
|
Theorem | fimacnvdisj 5307 |
The preimage of a class disjoint with a mapping's codomain is empty.
(Contributed by FL, 24-Jan-2007.)
|
⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐵 ∩ 𝐶) = ∅) → (◡𝐹 “ 𝐶) = ∅) |
|
Theorem | fintm 5308* |
Function into an intersection. (Contributed by Jim Kingdon,
28-Dec-2018.)
|
⊢ ∃𝑥 𝑥 ∈ 𝐵 ⇒ ⊢ (𝐹:𝐴⟶∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝐹:𝐴⟶𝑥) |
|
Theorem | fin 5309 |
Mapping into an intersection. (Contributed by NM, 14-Sep-1999.) (Proof
shortened by Andrew Salmon, 17-Sep-2011.)
|
⊢ (𝐹:𝐴⟶(𝐵 ∩ 𝐶) ↔ (𝐹:𝐴⟶𝐵 ∧ 𝐹:𝐴⟶𝐶)) |
|
Theorem | fabexg 5310* |
Existence of a set of functions. (Contributed by Paul Chapman,
25-Feb-2008.)
|
⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)} ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V) |
|
Theorem | fabex 5311* |
Existence of a set of functions. (Contributed by NM, 3-Dec-2007.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V & ⊢ 𝐹 = {𝑥 ∣ (𝑥:𝐴⟶𝐵 ∧ 𝜑)} ⇒ ⊢ 𝐹 ∈ V |
|
Theorem | dmfex 5312 |
If a mapping is a set, its domain is a set. (Contributed by NM,
27-Aug-2006.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
|
⊢ ((𝐹 ∈ 𝐶 ∧ 𝐹:𝐴⟶𝐵) → 𝐴 ∈ V) |
|
Theorem | f0 5313 |
The empty function. (Contributed by NM, 14-Aug-1999.)
|
⊢ ∅:∅⟶𝐴 |
|
Theorem | f00 5314 |
A class is a function with empty codomain iff it and its domain are empty.
(Contributed by NM, 10-Dec-2003.)
|
⊢ (𝐹:𝐴⟶∅ ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
|
Theorem | f0bi 5315 |
A function with empty domain is empty. (Contributed by Alexander van der
Vekens, 30-Jun-2018.)
|
⊢ (𝐹:∅⟶𝑋 ↔ 𝐹 = ∅) |
|
Theorem | f0dom0 5316 |
A function is empty iff it has an empty domain. (Contributed by AV,
10-Feb-2019.)
|
⊢ (𝐹:𝑋⟶𝑌 → (𝑋 = ∅ ↔ 𝐹 = ∅)) |
|
Theorem | f0rn0 5317* |
If there is no element in the range of a function, its domain must be
empty. (Contributed by Alexander van der Vekens, 12-Jul-2018.)
|
⊢ ((𝐸:𝑋⟶𝑌 ∧ ¬ ∃𝑦 ∈ 𝑌 𝑦 ∈ ran 𝐸) → 𝑋 = ∅) |
|
Theorem | fconst 5318 |
A cross product with a singleton is a constant function. (Contributed
by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon,
17-Sep-2011.)
|
⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴 × {𝐵}):𝐴⟶{𝐵} |
|
Theorem | fconstg 5319 |
A cross product with a singleton is a constant function. (Contributed
by NM, 19-Oct-2004.)
|
⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) |
|
Theorem | fnconstg 5320 |
A cross product with a singleton is a constant function. (Contributed by
NM, 24-Jul-2014.)
|
⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}) Fn 𝐴) |
|
Theorem | fconst6g 5321 |
Constant function with loose range. (Contributed by Stefan O'Rear,
1-Feb-2015.)
|
⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶𝐶) |
|
Theorem | fconst6 5322 |
A constant function as a mapping. (Contributed by Jeff Madsen,
30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)
|
⊢ 𝐵 ∈ 𝐶 ⇒ ⊢ (𝐴 × {𝐵}):𝐴⟶𝐶 |
|
Theorem | f1eq1 5323 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
|
⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐵 ↔ 𝐺:𝐴–1-1→𝐵)) |
|
Theorem | f1eq2 5324 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
|
⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐹:𝐵–1-1→𝐶)) |
|
Theorem | f1eq3 5325 |
Equality theorem for one-to-one functions. (Contributed by NM,
10-Feb-1997.)
|
⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1→𝐴 ↔ 𝐹:𝐶–1-1→𝐵)) |
|
Theorem | nff1 5326 |
Bound-variable hypothesis builder for a one-to-one function.
(Contributed by NM, 16-May-2004.)
|
⊢ Ⅎ𝑥𝐹
& ⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐹:𝐴–1-1→𝐵 |
|
Theorem | dff12 5327* |
Alternate definition of a one-to-one function. (Contributed by NM,
31-Dec-1996.)
|
⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦∃*𝑥 𝑥𝐹𝑦)) |
|
Theorem | f1f 5328 |
A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
|
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) |
|
Theorem | f1rn 5329 |
The range of a one-to-one mapping. (Contributed by BJ, 6-Jul-2022.)
|
⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵) |
|
Theorem | f1fn 5330 |
A one-to-one mapping is a function on its domain. (Contributed by NM,
8-Mar-2014.)
|
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴) |
|
Theorem | f1fun 5331 |
A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
|
⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹) |
|
Theorem | f1rel 5332 |
A one-to-one onto mapping is a relation. (Contributed by NM,
8-Mar-2014.)
|
⊢ (𝐹:𝐴–1-1→𝐵 → Rel 𝐹) |
|
Theorem | f1dm 5333 |
The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
|
⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴) |
|
Theorem | f1ss 5334 |
A function that is one-to-one is also one-to-one on some superset of its
range. (Contributed by Mario Carneiro, 12-Jan-2013.)
|
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶) |
|
Theorem | f1ssr 5335 |
Combine a one-to-one function with a restriction on the domain.
(Contributed by Stefan O'Rear, 20-Feb-2015.)
|
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 ⊆ 𝐶) → 𝐹:𝐴–1-1→𝐶) |
|
Theorem | f1ff1 5336 |
If a function is one-to-one from A to B and is also a function from A to
C, then it is a one-to-one function from A to C. (Contributed by BJ,
4-Jul-2022.)
|
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐹:𝐴⟶𝐶) → 𝐹:𝐴–1-1→𝐶) |
|
Theorem | f1ssres 5337 |
A function that is one-to-one is also one-to-one on any subclass of its
domain. (Contributed by Mario Carneiro, 17-Jan-2015.)
|
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐵) |
|
Theorem | f1resf1 5338 |
The restriction of an injective function is injective. (Contributed by
AV, 28-Jun-2022.)
|
⊢ (((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) ∧ (𝐹 ↾ 𝐶):𝐶⟶𝐷) → (𝐹 ↾ 𝐶):𝐶–1-1→𝐷) |
|
Theorem | f1cnvcnv 5339 |
Two ways to express that a set 𝐴 (not necessarily a function) is
one-to-one. Each side is equivalent to Definition 6.4(3) of
[TakeutiZaring] p. 24, who use the
notation "Un2 (A)" for one-to-one.
We
do not introduce a separate notation since we rarely use it. (Contributed
by NM, 13-Aug-2004.)
|
⊢ (◡◡𝐴:dom 𝐴–1-1→V ↔ (Fun ◡𝐴 ∧ Fun ◡◡𝐴)) |
|
Theorem | f1co 5340 |
Composition of one-to-one functions. Exercise 30 of [TakeutiZaring]
p. 25. (Contributed by NM, 28-May-1998.)
|
⊢ ((𝐹:𝐵–1-1→𝐶 ∧ 𝐺:𝐴–1-1→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1→𝐶) |
|
Theorem | foeq1 5341 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|
⊢ (𝐹 = 𝐺 → (𝐹:𝐴–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵)) |
|
Theorem | foeq2 5342 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|
⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶)) |
|
Theorem | foeq3 5343 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|
⊢ (𝐴 = 𝐵 → (𝐹:𝐶–onto→𝐴 ↔ 𝐹:𝐶–onto→𝐵)) |
|
Theorem | nffo 5344 |
Bound-variable hypothesis builder for an onto function. (Contributed by
NM, 16-May-2004.)
|
⊢ Ⅎ𝑥𝐹
& ⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵 |
|
Theorem | fof 5345 |
An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
|
⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) |
|
Theorem | fofun 5346 |
An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
|
⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹) |
|
Theorem | fofn 5347 |
An onto mapping is a function on its domain. (Contributed by NM,
16-Dec-2008.)
|
⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) |
|
Theorem | forn 5348 |
The codomain of an onto function is its range. (Contributed by NM,
3-Aug-1994.)
|
⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) |
|
Theorem | dffo2 5349 |
Alternate definition of an onto function. (Contributed by NM,
22-Mar-2006.)
|
⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) |
|
Theorem | foima 5350 |
The image of the domain of an onto function. (Contributed by NM,
29-Nov-2002.)
|
⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵) |
|
Theorem | dffn4 5351 |
A function maps onto its range. (Contributed by NM, 10-May-1998.)
|
⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) |
|
Theorem | funforn 5352 |
A function maps its domain onto its range. (Contributed by NM,
23-Jul-2004.)
|
⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴) |
|
Theorem | fodmrnu 5353 |
An onto function has unique domain and range. (Contributed by NM,
5-Nov-2006.)
|
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐹:𝐶–onto→𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
|
Theorem | fores 5354 |
Restriction of a function. (Contributed by NM, 4-Mar-1997.)
|
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) |
|
Theorem | foco 5355 |
Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
|
⊢ ((𝐹:𝐵–onto→𝐶 ∧ 𝐺:𝐴–onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–onto→𝐶) |
|
Theorem | f1oeq1 5356 |
Equality theorem for one-to-one onto functions. (Contributed by NM,
10-Feb-1997.)
|
⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵)) |
|
Theorem | f1oeq2 5357 |
Equality theorem for one-to-one onto functions. (Contributed by NM,
10-Feb-1997.)
|
⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶)) |
|
Theorem | f1oeq3 5358 |
Equality theorem for one-to-one onto functions. (Contributed by NM,
10-Feb-1997.)
|
⊢ (𝐴 = 𝐵 → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:𝐶–1-1-onto→𝐵)) |
|
Theorem | f1oeq23 5359 |
Equality theorem for one-to-one onto functions. (Contributed by FL,
14-Jul-2012.)
|
⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐷)) |
|
Theorem | f1eq123d 5360 |
Equality deduction for one-to-one functions. (Contributed by Mario
Carneiro, 27-Jan-2017.)
|
⊢ (𝜑 → 𝐹 = 𝐺)
& ⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐺:𝐵–1-1→𝐷)) |
|
Theorem | foeq123d 5361 |
Equality deduction for onto functions. (Contributed by Mario Carneiro,
27-Jan-2017.)
|
⊢ (𝜑 → 𝐹 = 𝐺)
& ⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐹:𝐴–onto→𝐶 ↔ 𝐺:𝐵–onto→𝐷)) |
|
Theorem | f1oeq123d 5362 |
Equality deduction for one-to-one onto functions. (Contributed by Mario
Carneiro, 27-Jan-2017.)
|
⊢ (𝜑 → 𝐹 = 𝐺)
& ⊢ (𝜑 → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝐶 = 𝐷) ⇒ ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐺:𝐵–1-1-onto→𝐷)) |
|
Theorem | f1oeq2d 5363 |
Equality deduction for one-to-one onto functions. (Contributed by
Glauco Siliprandi, 17-Aug-2020.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶)) |
|
Theorem | f1oeq3d 5364 |
Equality deduction for one-to-one onto functions. (Contributed by
Glauco Siliprandi, 17-Aug-2020.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:𝐶–1-1-onto→𝐵)) |
|
Theorem | nff1o 5365 |
Bound-variable hypothesis builder for a one-to-one onto function.
(Contributed by NM, 16-May-2004.)
|
⊢ Ⅎ𝑥𝐹
& ⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐹:𝐴–1-1-onto→𝐵 |
|
Theorem | f1of1 5366 |
A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM,
12-Dec-2003.)
|
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–1-1→𝐵) |
|
Theorem | f1of 5367 |
A one-to-one onto mapping is a mapping. (Contributed by NM,
12-Dec-2003.)
|
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵) |
|
Theorem | f1ofn 5368 |
A one-to-one onto mapping is function on its domain. (Contributed by NM,
12-Dec-2003.)
|
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴) |
|
Theorem | f1ofun 5369 |
A one-to-one onto mapping is a function. (Contributed by NM,
12-Dec-2003.)
|
⊢ (𝐹:𝐴–1-1-onto→𝐵 → Fun 𝐹) |
|
Theorem | f1orel 5370 |
A one-to-one onto mapping is a relation. (Contributed by NM,
13-Dec-2003.)
|
⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹) |
|
Theorem | f1odm 5371 |
The domain of a one-to-one onto mapping. (Contributed by NM,
8-Mar-2014.)
|
⊢ (𝐹:𝐴–1-1-onto→𝐵 → dom 𝐹 = 𝐴) |
|
Theorem | dff1o2 5372 |
Alternate definition of one-to-one onto function. (Contributed by NM,
10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵)) |
|
Theorem | dff1o3 5373 |
Alternate definition of one-to-one onto function. (Contributed by NM,
25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–onto→𝐵 ∧ Fun ◡𝐹)) |
|
Theorem | f1ofo 5374 |
A one-to-one onto function is an onto function. (Contributed by NM,
28-Apr-2004.)
|
⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴–onto→𝐵) |
|
Theorem | dff1o4 5375 |
Alternate definition of one-to-one onto function. (Contributed by NM,
25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵)) |
|
Theorem | dff1o5 5376 |
Alternate definition of one-to-one onto function. (Contributed by NM,
10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
|
⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹:𝐴–1-1→𝐵 ∧ ran 𝐹 = 𝐵)) |
|
Theorem | f1orn 5377 |
A one-to-one function maps onto its range. (Contributed by NM,
13-Aug-2004.)
|
⊢ (𝐹:𝐴–1-1-onto→ran
𝐹 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹)) |
|
Theorem | f1f1orn 5378 |
A one-to-one function maps one-to-one onto its range. (Contributed by NM,
4-Sep-2004.)
|
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran
𝐹) |
|
Theorem | f1oabexg 5379* |
The class of all 1-1-onto functions mapping one set to another is a set.
(Contributed by Paul Chapman, 25-Feb-2008.)
|
⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐵 ∧ 𝜑)} ⇒ ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → 𝐹 ∈ V) |
|
Theorem | f1ocnv 5380 |
The converse of a one-to-one onto function is also one-to-one onto.
(Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon,
22-Oct-2011.)
|
⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴) |
|
Theorem | f1ocnvb 5381 |
A relation is a one-to-one onto function iff its converse is a one-to-one
onto function with domain and range interchanged. (Contributed by NM,
8-Dec-2003.)
|
⊢ (Rel 𝐹 → (𝐹:𝐴–1-1-onto→𝐵 ↔ ◡𝐹:𝐵–1-1-onto→𝐴)) |
|
Theorem | f1ores 5382 |
The restriction of a one-to-one function maps one-to-one onto the image.
(Contributed by NM, 25-Mar-1998.)
|
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐹 ↾ 𝐶):𝐶–1-1-onto→(𝐹 “ 𝐶)) |
|
Theorem | f1orescnv 5383 |
The converse of a one-to-one-onto restricted function. (Contributed by
Paul Chapman, 21-Apr-2008.)
|
⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → (◡𝐹 ↾ 𝑃):𝑃–1-1-onto→𝑅) |
|
Theorem | f1imacnv 5384 |
Preimage of an image. (Contributed by NM, 30-Sep-2004.)
|
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝐶)) = 𝐶) |
|
Theorem | foimacnv 5385 |
A reverse version of f1imacnv 5384. (Contributed by Jeff Hankins,
16-Jul-2009.)
|
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝐶)) = 𝐶) |
|
Theorem | foun 5386 |
The union of two onto functions with disjoint domains is an onto function.
(Contributed by Mario Carneiro, 22-Jun-2016.)
|
⊢ (((𝐹:𝐴–onto→𝐵 ∧ 𝐺:𝐶–onto→𝐷) ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–onto→(𝐵 ∪ 𝐷)) |
|
Theorem | f1oun 5387 |
The union of two one-to-one onto functions with disjoint domains and
ranges. (Contributed by NM, 26-Mar-1998.)
|
⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷)) |
|
Theorem | fun11iun 5388* |
The union of a chain (with respect to inclusion) of one-to-one functions
is a one-to-one function. (Contributed by Mario Carneiro, 20-May-2013.)
(Revised by Mario Carneiro, 24-Jun-2015.)
|
⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
& ⊢ 𝐵 ∈ V ⇒ ⊢ (∀𝑥 ∈ 𝐴 (𝐵:𝐷–1-1→𝑆 ∧ ∀𝑦 ∈ 𝐴 (𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵)) → ∪ 𝑥 ∈ 𝐴 𝐵:∪ 𝑥 ∈ 𝐴 𝐷–1-1→𝑆) |
|
Theorem | resdif 5389 |
The restriction of a one-to-one onto function to a difference maps onto
the difference of the images. (Contributed by Paul Chapman,
11-Apr-2009.)
|
⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝐴):𝐴–onto→𝐶 ∧ (𝐹 ↾ 𝐵):𝐵–onto→𝐷) → (𝐹 ↾ (𝐴 ∖ 𝐵)):(𝐴 ∖ 𝐵)–1-1-onto→(𝐶 ∖ 𝐷)) |
|
Theorem | f1oco 5390 |
Composition of one-to-one onto functions. (Contributed by NM,
19-Mar-1998.)
|
⊢ ((𝐹:𝐵–1-1-onto→𝐶 ∧ 𝐺:𝐴–1-1-onto→𝐵) → (𝐹 ∘ 𝐺):𝐴–1-1-onto→𝐶) |
|
Theorem | f1cnv 5391 |
The converse of an injective function is bijective. (Contributed by FL,
11-Nov-2011.)
|
⊢ (𝐹:𝐴–1-1→𝐵 → ◡𝐹:ran 𝐹–1-1-onto→𝐴) |
|
Theorem | funcocnv2 5392 |
Composition with the converse. (Contributed by Jeff Madsen,
2-Sep-2009.)
|
⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
|
Theorem | fococnv2 5393 |
The composition of an onto function and its converse. (Contributed by
Stefan O'Rear, 12-Feb-2015.)
|
⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) |
|
Theorem | f1ococnv2 5394 |
The composition of a one-to-one onto function and its converse equals the
identity relation restricted to the function's range. (Contributed by NM,
13-Dec-2003.) (Proof shortened by Stefan O'Rear, 12-Feb-2015.)
|
⊢ (𝐹:𝐴–1-1-onto→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ 𝐵)) |
|
Theorem | f1cocnv2 5395 |
Composition of an injective function with its converse. (Contributed by
FL, 11-Nov-2011.)
|
⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
|
Theorem | f1ococnv1 5396 |
The composition of a one-to-one onto function's converse and itself equals
the identity relation restricted to the function's domain. (Contributed
by NM, 13-Dec-2003.)
|
⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) |
|
Theorem | f1cocnv1 5397 |
Composition of an injective function with its converse. (Contributed by
FL, 11-Nov-2011.)
|
⊢ (𝐹:𝐴–1-1→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) |
|
Theorem | funcoeqres 5398 |
Express a constraint on a composition as a constraint on the composand.
(Contributed by Stefan O'Rear, 7-Mar-2015.)
|
⊢ ((Fun 𝐺 ∧ (𝐹 ∘ 𝐺) = 𝐻) → (𝐹 ↾ ran 𝐺) = (𝐻 ∘ ◡𝐺)) |
|
Theorem | ffoss 5399* |
Relationship between a mapping and an onto mapping. Figure 38 of
[Enderton] p. 145. (Contributed by NM,
10-May-1998.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐹:𝐴⟶𝐵 ↔ ∃𝑥(𝐹:𝐴–onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |
|
Theorem | f11o 5400* |
Relationship between one-to-one and one-to-one onto function.
(Contributed by NM, 4-Apr-1998.)
|
⊢ 𝐹 ∈ V ⇒ ⊢ (𝐹:𝐴–1-1→𝐵 ↔ ∃𝑥(𝐹:𝐴–1-1-onto→𝑥 ∧ 𝑥 ⊆ 𝐵)) |