Theorem List for Intuitionistic Logic Explorer - 5501-5600 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | fun11iun 5501* |
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 5502 |
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 5503 |
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 5504 |
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 5505 |
Composition with the converse. (Contributed by Jeff Madsen,
2-Sep-2009.)
|
⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)) |
|
Theorem | fococnv2 5506 |
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 5507 |
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 5508 |
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 5509 |
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 5510 |
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 5511 |
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 5512* |
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 5513* |
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 5514 |
The empty set maps one-to-one into any class. (Contributed by NM,
7-Apr-1998.)
|
⊢ ∅:∅–1-1→𝐴 |
|
Theorem | f1o00 5515 |
One-to-one onto mapping of the empty set. (Contributed by NM,
15-Apr-1998.)
|
⊢ (𝐹:∅–1-1-onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
|
Theorem | fo00 5516 |
Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
|
⊢ (𝐹:∅–onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
|
Theorem | f1o0 5517 |
One-to-one onto mapping of the empty set. (Contributed by NM,
10-Sep-2004.)
|
⊢ ∅:∅–1-1-onto→∅ |
|
Theorem | f1oi 5518 |
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 5519 |
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 5520 |
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 5521 |
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 5522 |
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 5523 |
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 5524 |
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 5525* |
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 5526* |
The value of a function at a unique point. (Contributed by Scott
Fenton, 6-Oct-2017.)
|
⊢ (∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∪ {𝑥 ∣ 𝐴𝐹𝑥}) |
|
Theorem | brprcneu 5527* |
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 5528 |
A function's value at a proper class is the empty set. (Contributed by
NM, 20-May-1998.)
|
⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅) |
|
Theorem | fv2 5529* |
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 5530* |
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 5531* |
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 5015), 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 5532* |
Membership in a function value. (Contributed by NM, 30-Apr-2004.)
|
⊢ (𝐴 ∈ (𝐹‘𝐵) ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ ∀𝑦(𝐵𝐹𝑦 ↔ 𝑦 = 𝑥))) |
|
Theorem | fveq1 5533 |
Equality theorem for function value. (Contributed by NM,
29-Dec-1996.)
|
⊢ (𝐹 = 𝐺 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
|
Theorem | fveq2 5534 |
Equality theorem for function value. (Contributed by NM,
29-Dec-1996.)
|
⊢ (𝐴 = 𝐵 → (𝐹‘𝐴) = (𝐹‘𝐵)) |
|
Theorem | fveq1i 5535 |
Equality inference for function value. (Contributed by NM,
2-Sep-2003.)
|
⊢ 𝐹 = 𝐺 ⇒ ⊢ (𝐹‘𝐴) = (𝐺‘𝐴) |
|
Theorem | fveq1d 5536 |
Equality deduction for function value. (Contributed by NM,
2-Sep-2003.)
|
⊢ (𝜑 → 𝐹 = 𝐺) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴)) |
|
Theorem | fveq2i 5537 |
Equality inference for function value. (Contributed by NM,
28-Jul-1999.)
|
⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐹‘𝐴) = (𝐹‘𝐵) |
|
Theorem | fveq2d 5538 |
Equality deduction for function value. (Contributed by NM,
29-May-1999.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵)) |
|
Theorem | 2fveq3 5539 |
Equality theorem for nested function values. (Contributed by AV,
14-Aug-2022.)
|
⊢ (𝐴 = 𝐵 → (𝐹‘(𝐺‘𝐴)) = (𝐹‘(𝐺‘𝐵))) |
|
Theorem | fveq12i 5540 |
Equality deduction for function value. (Contributed by FL,
27-Jun-2014.)
|
⊢ 𝐹 = 𝐺
& ⊢ 𝐴 = 𝐵 ⇒ ⊢ (𝐹‘𝐴) = (𝐺‘𝐵) |
|
Theorem | fveq12d 5541 |
Equality deduction for function value. (Contributed by FL,
22-Dec-2008.)
|
⊢ (𝜑 → 𝐹 = 𝐺)
& ⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐵)) |
|
Theorem | fveqeq2d 5542 |
Equality deduction for function value. (Contributed by BJ,
30-Aug-2022.)
|
⊢ (𝜑 → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ((𝐹‘𝐴) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶)) |
|
Theorem | fveqeq2 5543 |
Equality deduction for function value. (Contributed by BJ,
31-Aug-2022.)
|
⊢ (𝐴 = 𝐵 → ((𝐹‘𝐴) = 𝐶 ↔ (𝐹‘𝐵) = 𝐶)) |
|
Theorem | nffv 5544 |
Bound-variable hypothesis builder for function value. (Contributed by
NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
|
⊢ Ⅎ𝑥𝐹
& ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ Ⅎ𝑥(𝐹‘𝐴) |
|
Theorem | nffvmpt1 5545* |
Bound-variable hypothesis builder for mapping, special case.
(Contributed by Mario Carneiro, 25-Dec-2016.)
|
⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝐶) |
|
Theorem | nffvd 5546 |
Deduction version of bound-variable hypothesis builder nffv 5544.
(Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro,
15-Oct-2016.)
|
⊢ (𝜑 → Ⅎ𝑥𝐹)
& ⊢ (𝜑 → Ⅎ𝑥𝐴) ⇒ ⊢ (𝜑 → Ⅎ𝑥(𝐹‘𝐴)) |
|
Theorem | funfveu 5547* |
A function has one value given an argument in its domain. (Contributed
by Jim Kingdon, 29-Dec-2018.)
|
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦) |
|
Theorem | fvss 5548* |
The value of a function is a subset of 𝐵 if every element that could
be a candidate for the value is a subset of 𝐵. (Contributed by
Mario Carneiro, 24-May-2019.)
|
⊢ (∀𝑥(𝐴𝐹𝑥 → 𝑥 ⊆ 𝐵) → (𝐹‘𝐴) ⊆ 𝐵) |
|
Theorem | fvssunirng 5549 |
The result of a function value is always a subset of the union of the
range, if the input is a set. (Contributed by Stefan O'Rear,
2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
|
⊢ (𝐴 ∈ V → (𝐹‘𝐴) ⊆ ∪ ran
𝐹) |
|
Theorem | relfvssunirn 5550 |
The result of a function value is always a subset of the union of the
range, even if it is invalid and thus empty. (Contributed by Stefan
O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
|
⊢ (Rel 𝐹 → (𝐹‘𝐴) ⊆ ∪ ran
𝐹) |
|
Theorem | funfvex 5551 |
The value of a function exists. A special case of Corollary 6.13 of
[TakeutiZaring] p. 27.
(Contributed by Jim Kingdon, 29-Dec-2018.)
|
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ V) |
|
Theorem | relrnfvex 5552 |
If a function has a set range, then the function value exists
unconditional on the domain. (Contributed by Mario Carneiro,
24-May-2019.)
|
⊢ ((Rel 𝐹 ∧ ran 𝐹 ∈ V) → (𝐹‘𝐴) ∈ V) |
|
Theorem | fvexg 5553 |
Evaluating a set function at a set exists. (Contributed by Mario
Carneiro and Jim Kingdon, 28-May-2019.)
|
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐹‘𝐴) ∈ V) |
|
Theorem | fvex 5554 |
Evaluating a set function at a set exists. (Contributed by Mario
Carneiro and Jim Kingdon, 28-May-2019.)
|
⊢ 𝐹 ∈ 𝑉
& ⊢ 𝐴 ∈ 𝑊 ⇒ ⊢ (𝐹‘𝐴) ∈ V |
|
Theorem | sefvex 5555 |
If a function is set-like, then the function value exists if the input
does. (Contributed by Mario Carneiro, 24-May-2019.)
|
⊢ ((◡𝐹 Se V ∧ 𝐴 ∈ V) → (𝐹‘𝐴) ∈ V) |
|
Theorem | fvifdc 5556 |
Move a conditional outside of a function. (Contributed by Jim Kingdon,
1-Jan-2022.)
|
⊢ (DECID 𝜑 → (𝐹‘if(𝜑, 𝐴, 𝐵)) = if(𝜑, (𝐹‘𝐴), (𝐹‘𝐵))) |
|
Theorem | fv3 5557* |
Alternate definition of the value of a function. Definition 6.11 of
[TakeutiZaring] p. 26.
(Contributed by NM, 30-Apr-2004.) (Revised by
Mario Carneiro, 31-Aug-2015.)
|
⊢ (𝐹‘𝐴) = {𝑥 ∣ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝐴𝐹𝑦) ∧ ∃!𝑦 𝐴𝐹𝑦)} |
|
Theorem | fvres 5558 |
The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
|
⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴)) |
|
Theorem | fvresd 5559 |
The value of a restricted function, deduction version of fvres 5558.
(Contributed by Glauco Siliprandi, 8-Apr-2021.)
|
⊢ (𝜑 → 𝐴 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴)) |
|
Theorem | funssfv 5560 |
The value of a member of the domain of a subclass of a function.
(Contributed by NM, 15-Aug-1994.)
|
⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴)) |
|
Theorem | tz6.12-1 5561* |
Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed
by NM, 30-Apr-2004.)
|
⊢ ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹‘𝐴) = 𝑦) |
|
Theorem | tz6.12 5562* |
Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed
by NM, 10-Jul-1994.)
|
⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑦) |
|
Theorem | tz6.12f 5563* |
Function value, using bound-variable hypotheses instead of distinct
variable conditions. (Contributed by NM, 30-Aug-1999.)
|
⊢ Ⅎ𝑦𝐹 ⇒ ⊢ ((〈𝐴, 𝑦〉 ∈ 𝐹 ∧ ∃!𝑦〈𝐴, 𝑦〉 ∈ 𝐹) → (𝐹‘𝐴) = 𝑦) |
|
Theorem | tz6.12c 5564* |
Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by
NM, 30-Apr-2004.)
|
⊢ (∃!𝑦 𝐴𝐹𝑦 → ((𝐹‘𝐴) = 𝑦 ↔ 𝐴𝐹𝑦)) |
|
Theorem | ndmfvg 5565 |
The value of a class outside its domain is the empty set. (Contributed
by Jim Kingdon, 15-Jan-2019.)
|
⊢ ((𝐴 ∈ V ∧ ¬ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∅) |
|
Theorem | relelfvdm 5566 |
If a function value has a member, the argument belongs to the domain.
(Contributed by Jim Kingdon, 22-Jan-2019.)
|
⊢ ((Rel 𝐹 ∧ 𝐴 ∈ (𝐹‘𝐵)) → 𝐵 ∈ dom 𝐹) |
|
Theorem | elfvm 5567* |
If a function value has a member, the function is inhabited.
(Contributed by Jim Kingdon, 14-Jun-2025.)
|
⊢ (𝐴 ∈ (𝐹‘𝐵) → ∃𝑗 𝑗 ∈ 𝐹) |
|
Theorem | nfvres 5568 |
The value of a non-member of a restriction is the empty set.
(Contributed by NM, 13-Nov-1995.)
|
⊢ (¬ 𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = ∅) |
|
Theorem | nfunsn 5569 |
If the restriction of a class to a singleton is not a function, its
value is the empty set. (Contributed by NM, 8-Aug-2010.) (Proof
shortened by Andrew Salmon, 22-Oct-2011.)
|
⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅) |
|
Theorem | 0fv 5570 |
Function value of the empty set. (Contributed by Stefan O'Rear,
26-Nov-2014.)
|
⊢ (∅‘𝐴) = ∅ |
|
Theorem | fv2prc 5571 |
A function value of a function value at a proper class is the empty set.
(Contributed by AV, 8-Apr-2021.)
|
⊢ (¬ 𝐴 ∈ V → ((𝐹‘𝐴)‘𝐵) = ∅) |
|
Theorem | csbfv12g 5572 |
Move class substitution in and out of a function value. (Contributed by
NM, 11-Nov-2005.)
|
⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)) |
|
Theorem | csbfv2g 5573* |
Move class substitution in and out of a function value. (Contributed by
NM, 10-Nov-2005.)
|
⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (𝐹‘⦋𝐴 / 𝑥⦌𝐵)) |
|
Theorem | csbfvg 5574* |
Substitution for a function value. (Contributed by NM, 1-Jan-2006.)
|
⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝑥) = (𝐹‘𝐴)) |
|
Theorem | funbrfv 5575 |
The second argument of a binary relation on a function is the function's
value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro,
28-Apr-2015.)
|
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹‘𝐴) = 𝐵)) |
|
Theorem | funopfv 5576 |
The second element in an ordered pair member of a function is the
function's value. (Contributed by NM, 19-Jul-1996.)
|
⊢ (Fun 𝐹 → (〈𝐴, 𝐵〉 ∈ 𝐹 → (𝐹‘𝐴) = 𝐵)) |
|
Theorem | fnbrfvb 5577 |
Equivalence of function value and binary relation. (Contributed by NM,
19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
|
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ 𝐵𝐹𝐶)) |
|
Theorem | fnopfvb 5578 |
Equivalence of function value and ordered pair membership. (Contributed
by NM, 7-Nov-1995.)
|
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → ((𝐹‘𝐵) = 𝐶 ↔ 〈𝐵, 𝐶〉 ∈ 𝐹)) |
|
Theorem | funbrfvb 5579 |
Equivalence of function value and binary relation. (Contributed by NM,
26-Mar-2006.)
|
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) = 𝐵 ↔ 𝐴𝐹𝐵)) |
|
Theorem | funopfvb 5580 |
Equivalence of function value and ordered pair membership. Theorem
4.3(ii) of [Monk1] p. 42. (Contributed by
NM, 26-Jan-1997.)
|
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ((𝐹‘𝐴) = 𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝐹)) |
|
Theorem | funbrfv2b 5581 |
Function value in terms of a binary relation. (Contributed by Mario
Carneiro, 19-Mar-2014.)
|
⊢ (Fun 𝐹 → (𝐴𝐹𝐵 ↔ (𝐴 ∈ dom 𝐹 ∧ (𝐹‘𝐴) = 𝐵))) |
|
Theorem | dffn5im 5582* |
Representation of a function in terms of its values. The converse holds
given the law of the excluded middle; as it is we have most of the
converse via funmpt 5273 and dmmptss 5143. (Contributed by Jim Kingdon,
31-Dec-2018.)
|
⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
|
Theorem | fnrnfv 5583* |
The range of a function expressed as a collection of the function's
values. (Contributed by NM, 20-Oct-2005.) (Proof shortened by Mario
Carneiro, 31-Aug-2015.)
|
⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) |
|
Theorem | fvelrnb 5584* |
A member of a function's range is a value of the function. (Contributed
by NM, 31-Oct-1995.)
|
⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵)) |
|
Theorem | dfimafn 5585* |
Alternate definition of the image of a function. (Contributed by Raph
Levien, 20-Nov-2006.)
|
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦}) |
|
Theorem | dfimafn2 5586* |
Alternate definition of the image of a function as an indexed union of
singletons of function values. (Contributed by Raph Levien,
20-Nov-2006.)
|
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = ∪
𝑥 ∈ 𝐴 {(𝐹‘𝑥)}) |
|
Theorem | funimass4 5587* |
Membership relation for the values of a function whose image is a
subclass. (Contributed by Raph Levien, 20-Nov-2006.)
|
⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
|
Theorem | fvelima 5588* |
Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42.
(Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon,
22-Oct-2011.)
|
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (𝐹 “ 𝐵)) → ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐴) |
|
Theorem | foelcdmi 5589* |
A member of a surjective function's codomain is a value of the function.
(Contributed by Thierry Arnoux, 23-Jan-2020.)
|
⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑌 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌) |
|
Theorem | feqmptd 5590* |
Deduction form of dffn5im 5582. (Contributed by Mario Carneiro,
8-Jan-2015.)
|
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
|
Theorem | feqresmpt 5591* |
Express a restricted function as a mapping. (Contributed by Mario
Carneiro, 18-May-2016.)
|
⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
& ⊢ (𝜑 → 𝐶 ⊆ 𝐴) ⇒ ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))) |
|
Theorem | dffn5imf 5592* |
Representation of a function in terms of its values. (Contributed by
Jim Kingdon, 31-Dec-2018.)
|
⊢ Ⅎ𝑥𝐹 ⇒ ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
|
Theorem | fvelimab 5593* |
Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof
shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy,
17-Dec-2011.)
|
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)) |
|
Theorem | fvi 5594 |
The value of the identity function. (Contributed by NM, 1-May-2004.)
(Revised by Mario Carneiro, 28-Apr-2015.)
|
⊢ (𝐴 ∈ 𝑉 → ( I ‘𝐴) = 𝐴) |
|
Theorem | fniinfv 5595* |
The indexed intersection of a function's values is the intersection of
its range. (Contributed by NM, 20-Oct-2005.)
|
⊢ (𝐹 Fn 𝐴 → ∩
𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∩ ran 𝐹) |
|
Theorem | fnsnfv 5596 |
Singleton of function value. (Contributed by NM, 22-May-1998.)
|
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴) → {(𝐹‘𝐵)} = (𝐹 “ {𝐵})) |
|
Theorem | fnimapr 5597 |
The image of a pair under a function. (Contributed by Jeff Madsen,
6-Jan-2011.)
|
⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (𝐹 “ {𝐵, 𝐶}) = {(𝐹‘𝐵), (𝐹‘𝐶)}) |
|
Theorem | ssimaex 5598* |
The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
|
⊢ 𝐴 ∈ V ⇒ ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))) |
|
Theorem | ssimaexg 5599* |
The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
|
⊢ ((𝐴 ∈ 𝐶 ∧ Fun 𝐹 ∧ 𝐵 ⊆ (𝐹 “ 𝐴)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝐵 = (𝐹 “ 𝑥))) |
|
Theorem | funfvdm 5600 |
A simplified expression for the value of a function when we know it's a
function. (Contributed by Jim Kingdon, 1-Jan-2019.)
|
⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴})) |