Theorem List for Intuitionistic Logic Explorer - 5401-5500 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | nff1 5401 |
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 5402* |
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 5403 |
A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
|
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) |
|
Theorem | f1rn 5404 |
The range of a one-to-one mapping. (Contributed by BJ, 6-Jul-2022.)
|
⊢ (𝐹:𝐴–1-1→𝐵 → ran 𝐹 ⊆ 𝐵) |
|
Theorem | f1fn 5405 |
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 5406 |
A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
|
⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹) |
|
Theorem | f1rel 5407 |
A one-to-one onto mapping is a relation. (Contributed by NM,
8-Mar-2014.)
|
⊢ (𝐹:𝐴–1-1→𝐵 → Rel 𝐹) |
|
Theorem | f1dm 5408 |
The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
|
⊢ (𝐹:𝐴–1-1→𝐵 → dom 𝐹 = 𝐴) |
|
Theorem | f1ss 5409 |
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 5410 |
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 5411 |
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 5412 |
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 5413 |
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 5414 |
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 5415 |
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 5416 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|
⊢ (𝐹 = 𝐺 → (𝐹:𝐴–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵)) |
|
Theorem | foeq2 5417 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|
⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶)) |
|
Theorem | foeq3 5418 |
Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
|
⊢ (𝐴 = 𝐵 → (𝐹:𝐶–onto→𝐴 ↔ 𝐹:𝐶–onto→𝐵)) |
|
Theorem | nffo 5419 |
Bound-variable hypothesis builder for an onto function. (Contributed by
NM, 16-May-2004.)
|
⊢ Ⅎ𝑥𝐹
& ⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ Ⅎ𝑥 𝐹:𝐴–onto→𝐵 |
|
Theorem | fof 5420 |
An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
|
⊢ (𝐹:𝐴–onto→𝐵 → 𝐹:𝐴⟶𝐵) |
|
Theorem | fofun 5421 |
An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
|
⊢ (𝐹:𝐴–onto→𝐵 → Fun 𝐹) |
|
Theorem | fofn 5422 |
An onto mapping is a function on its domain. (Contributed by NM,
16-Dec-2008.)
|
⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴) |
|
Theorem | forn 5423 |
The codomain of an onto function is its range. (Contributed by NM,
3-Aug-1994.)
|
⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵) |
|
Theorem | dffo2 5424 |
Alternate definition of an onto function. (Contributed by NM,
22-Mar-2006.)
|
⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ran 𝐹 = 𝐵)) |
|
Theorem | foima 5425 |
The image of the domain of an onto function. (Contributed by NM,
29-Nov-2002.)
|
⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵) |
|
Theorem | dffn4 5426 |
A function maps onto its range. (Contributed by NM, 10-May-1998.)
|
⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹) |
|
Theorem | funforn 5427 |
A function maps its domain onto its range. (Contributed by NM,
23-Jul-2004.)
|
⊢ (Fun 𝐴 ↔ 𝐴:dom 𝐴–onto→ran 𝐴) |
|
Theorem | fodmrnu 5428 |
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 5429 |
Restriction of a function. (Contributed by NM, 4-Mar-1997.)
|
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 ↾ 𝐴):𝐴–onto→(𝐹 “ 𝐴)) |
|
Theorem | foco 5430 |
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 5431 |
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 5432 |
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 5433 |
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 5434 |
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 5435 |
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 5436 |
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 5437 |
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 5438 |
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 5439 |
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 5440 |
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 5441 |
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 5442 |
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 5443 |
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 5444 |
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 5445 |
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 5446 |
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 5447 |
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 5448 |
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 5449 |
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 5450 |
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 5451 |
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 5452 |
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 5453 |
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 5454* |
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 5455 |
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 5456 |
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 5457 |
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 5458 |
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 5459 |
Preimage of an image. (Contributed by NM, 30-Sep-2004.)
|
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (◡𝐹 “ (𝐹 “ 𝐶)) = 𝐶) |
|
Theorem | foimacnv 5460 |
A reverse version of f1imacnv 5459. (Contributed by Jeff Hankins,
16-Jul-2009.)
|
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ⊆ 𝐵) → (𝐹 “ (◡𝐹 “ 𝐶)) = 𝐶) |
|
Theorem | foun 5461 |
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 5462 |
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 5463* |
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 5464 |
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 5465 |
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 5466 |
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 5467 |
Composition with the converse. (Contributed by Jeff Madsen,
2-Sep-2009.)
|
⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
|
Theorem | fococnv2 5468 |
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 5469 |
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 5470 |
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 5471 |
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 5472 |
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 5473 |
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 5474* |
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 5475* |
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)) |
|
Theorem | f10 5476 |
The empty set maps one-to-one into any class. (Contributed by NM,
7-Apr-1998.)
|
⊢ ∅:∅–1-1→𝐴 |
|
Theorem | f1o00 5477 |
One-to-one onto mapping of the empty set. (Contributed by NM,
15-Apr-1998.)
|
⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
|
Theorem | fo00 5478 |
Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
|
⊢ (𝐹:∅–onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
|
Theorem | f1o0 5479 |
One-to-one onto mapping of the empty set. (Contributed by NM,
10-Sep-2004.)
|
⊢ ∅:∅–1-1-onto→∅ |
|
Theorem | f1oi 5480 |
A restriction of the identity relation is a one-to-one onto function.
(Contributed by NM, 30-Apr-1998.) (Proof shortened by Andrew Salmon,
22-Oct-2011.)
|
⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 |
|
Theorem | f1ovi 5481 |
The identity relation is a one-to-one onto function on the universe.
(Contributed by NM, 16-May-2004.)
|
⊢ I :V–1-1-onto→V |
|
Theorem | f1osn 5482 |
A singleton of an ordered pair is one-to-one onto function.
(Contributed by NM, 18-May-1998.) (Proof shortened by Andrew Salmon,
22-Oct-2011.)
|
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈
V ⇒ ⊢ {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵} |
|
Theorem | f1osng 5483 |
A singleton of an ordered pair is one-to-one onto function.
(Contributed by Mario Carneiro, 12-Jan-2013.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1-onto→{𝐵}) |
|
Theorem | f1sng 5484 |
A singleton of an ordered pair is a one-to-one function. (Contributed
by AV, 17-Apr-2021.)
|
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉}:{𝐴}–1-1→𝑊) |
|
Theorem | fsnd 5485 |
A singleton of an ordered pair is a function. (Contributed by AV,
17-Apr-2021.)
|
⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐵 ∈ 𝑊) ⇒ ⊢ (𝜑 → {〈𝐴, 𝐵〉}:{𝐴}⟶𝑊) |
|
Theorem | f1oprg 5486 |
An unordered pair of ordered pairs with different elements is a one-to-one
onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
|
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌)) → ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷) → {〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉}:{𝐴, 𝐶}–1-1-onto→{𝐵, 𝐷})) |
|
Theorem | tz6.12-2 5487* |
Function value when 𝐹 is not a function. Theorem 6.12(2)
of
[TakeutiZaring] p. 27.
(Contributed by NM, 30-Apr-2004.) (Proof
shortened by Mario Carneiro, 31-Aug-2015.)
|
⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅) |
|
Theorem | fveu 5488* |
The value of a function at a unique point. (Contributed by Scott
Fenton, 6-Oct-2017.)
|
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}) |
|
Theorem | brprcneu 5489* |
If 𝐴 is a proper class and 𝐹 is any
class, then there is no
unique set which is related to 𝐴 through the binary relation 𝐹.
(Contributed by Scott Fenton, 7-Oct-2017.)
|
⊢ (¬ 𝐴 ∈ V → ¬ ∃!𝑥 𝐴𝐹𝑥) |
|
Theorem | fvprc 5490 |
A function's value at a proper class is the empty set. (Contributed by
NM, 20-May-1998.)
|
⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) |
|
Theorem | fv2 5491* |
Alternate definition of function value. Definition 10.11 of [Quine]
p. 68. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew
Salmon, 17-Sep-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
|
⊢ (𝐹‘𝐴) = ∪ {𝑥 ∣ ∀𝑦(𝐴𝐹𝑦 ↔ 𝑦 = 𝑥)} |
|
Theorem | dffv3g 5492* |
A definition of function value in terms of iota. (Contributed by Jim
Kingdon, 29-Dec-2018.)
|
⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))) |
|
Theorem | dffv4g 5493* |
The previous definition of function value, from before the ℩
operator was introduced. Although based on the idea embodied by
Definition 10.2 of [Quine] p. 65 (see args 4980), this definition
apparently does not appear in the literature. (Contributed by NM,
1-Aug-1994.)
|
⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = ∪ {𝑥 ∣ (𝐹 “ {𝐴}) = {𝑥}}) |
|
Theorem | elfv 5494* |
Membership in a function value. (Contributed by NM, 30-Apr-2004.)
|
⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥))) |
|
Theorem | fveq1 5495 |
Equality theorem for function value. (Contributed by NM,
29-Dec-1996.)
|
⊢ (𝐹 = 𝐺 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
|
Theorem | fveq2 5496 |
Equality theorem for function value. (Contributed by NM,
29-Dec-1996.)
|
⊢ (𝐴 = 𝐵 → (𝐹‘𝐴) = (𝐹‘𝐵)) |
|
Theorem | fveq1i 5497 |
Equality inference for function value. (Contributed by NM,
2-Sep-2003.)
|
⊢ 𝐹 = 𝐺 ⇒ ⊢ (𝐹‘𝐴) = (𝐺‘𝐴) |
|
Theorem | fveq1d 5498 |
Equality deduction for function value. (Contributed by NM,
2-Sep-2003.)
|
⊢ (𝜑 → 𝐹 = 𝐺) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
|
Theorem | fveq2i 5499 |
Equality inference for function value. (Contributed by NM,
28-Jul-1999.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐹‘𝐴) = (𝐹‘𝐵) |
|
Theorem | fveq2d 5500 |
Equality deduction for function value. (Contributed by NM,
29-May-1999.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵)) |