Theorem List for Intuitionistic Logic Explorer - 5401-5500 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
|
Theorem | fococnv2 5401 |
The composition of an onto function and its converse. (Contributed by
Stefan O'Rear, 12-Feb-2015.)
|
![( (](lp.gif) ![F F](_cf.gif) ![: :](colon.gif) ![A A](_ca.gif) ![-onto-> -onto->](onto.gif) ![( (](lp.gif) ![`' `'](_cnv.gif) ![F F](_cf.gif)
![B B](_cb.gif) ![) )](rp.gif) ![)
)](rp.gif) |
|
Theorem | f1ococnv2 5402 |
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.)
|
![( (](lp.gif) ![F F](_cf.gif) ![: :](colon.gif) ![A A](_ca.gif) ![-1-1-onto-> -1-1-onto->](onetooneonto.gif) ![( (](lp.gif) ![`' `'](_cnv.gif) ![F F](_cf.gif)
![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | f1cocnv2 5403 |
Composition of an injective function with its converse. (Contributed by
FL, 11-Nov-2011.)
|
![( (](lp.gif) ![F F](_cf.gif) ![: :](colon.gif) ![A A](_ca.gif) ![-1-1-> -1-1->](onetoone.gif) ![( (](lp.gif) ![`' `'](_cnv.gif) ![F F](_cf.gif)
![F F](_cf.gif) ![) )](rp.gif) ![)
)](rp.gif) |
|
Theorem | f1ococnv1 5404 |
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.)
|
![( (](lp.gif) ![F F](_cf.gif) ![: :](colon.gif) ![A A](_ca.gif) ![-1-1-onto-> -1-1-onto->](onetooneonto.gif) ![( (](lp.gif) ![`' `'](_cnv.gif) ![F F](_cf.gif)
![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | f1cocnv1 5405 |
Composition of an injective function with its converse. (Contributed by
FL, 11-Nov-2011.)
|
![( (](lp.gif) ![F F](_cf.gif) ![: :](colon.gif) ![A A](_ca.gif) ![-1-1-> -1-1->](onetoone.gif) ![( (](lp.gif) ![`' `'](_cnv.gif) ![F F](_cf.gif)
![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | funcoeqres 5406 |
Express a constraint on a composition as a constraint on the composand.
(Contributed by Stefan O'Rear, 7-Mar-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![G G](_cg.gif)
![H H](_ch.gif) ![( (](lp.gif) ![G G](_cg.gif)
![( (](lp.gif) ![`' `'](_cnv.gif) ![G G](_cg.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ffoss 5407* |
Relationship between a mapping and an onto mapping. Figure 38 of
[Enderton] p. 145. (Contributed by NM,
10-May-1998.)
|
![( (](lp.gif) ![F F](_cf.gif) ![: :](colon.gif) ![A A](_ca.gif) ![-->
-->](longrightarrow.gif) ![E. E.](exists.gif) ![x x](_x.gif) ![( (](lp.gif) ![F F](_cf.gif) ![: :](colon.gif) ![A A](_ca.gif) ![-onto-> -onto->](onto.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | f11o 5408* |
Relationship between one-to-one and one-to-one onto function.
(Contributed by NM, 4-Apr-1998.)
|
![( (](lp.gif) ![F F](_cf.gif) ![: :](colon.gif) ![A A](_ca.gif) ![-1-1-> -1-1->](onetoone.gif)
![E. E.](exists.gif) ![x x](_x.gif) ![( (](lp.gif) ![F F](_cf.gif) ![: :](colon.gif) ![A A](_ca.gif) ![-1-1-onto-> -1-1-onto->](onetooneonto.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | f10 5409 |
The empty set maps one-to-one into any class. (Contributed by NM,
7-Apr-1998.)
|
![(/) (/)](varnothing.gif) ![: :](colon.gif) ![(/) (/)](varnothing.gif) ![-1-1-> -1-1->](onetoone.gif) ![A A](_ca.gif) |
|
Theorem | f1o00 5410 |
One-to-one onto mapping of the empty set. (Contributed by NM,
15-Apr-1998.)
|
![( (](lp.gif) ![F F](_cf.gif) ![: :](colon.gif) ![(/) (/)](varnothing.gif) ![-1-1-onto-> -1-1-onto->](onetooneonto.gif) ![( (](lp.gif) ![(/) (/)](varnothing.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fo00 5411 |
Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
|
![( (](lp.gif) ![F F](_cf.gif) ![: :](colon.gif) ![(/) (/)](varnothing.gif) ![-onto-> -onto->](onto.gif) ![(
(](lp.gif) ![(/) (/)](varnothing.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | f1o0 5412 |
One-to-one onto mapping of the empty set. (Contributed by NM,
10-Sep-2004.)
|
![(/) (/)](varnothing.gif) ![: :](colon.gif) ![(/) (/)](varnothing.gif) ![-1-1-onto->
-1-1-onto->](onetooneonto.gif) ![(/) (/)](varnothing.gif) |
|
Theorem | f1oi 5413 |
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.)
|
![A A](_ca.gif) ![) )](rp.gif) ![: :](colon.gif) ![A A](_ca.gif) ![-1-1-onto-> -1-1-onto->](onetooneonto.gif) ![A A](_ca.gif) |
|
Theorem | f1ovi 5414 |
The identity relation is a one-to-one onto function on the universe.
(Contributed by NM, 16-May-2004.)
|
![: :](colon.gif) ![_V _V](rmcv.gif) ![-1-1-onto-> -1-1-onto->](onetooneonto.gif) ![_V _V](rmcv.gif) |
|
Theorem | f1osn 5415 |
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.)
|
![{
{](lbrace.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![>. >.](rangle.gif) ![} }](rbrace.gif) ![: :](colon.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![} }](rbrace.gif) ![-1-1-onto->
-1-1-onto->](onetooneonto.gif) ![{ {](lbrace.gif) ![B B](_cb.gif) ![} }](rbrace.gif) |
|
Theorem | f1osng 5416 |
A singleton of an ordered pair is one-to-one onto function.
(Contributed by Mario Carneiro, 12-Jan-2013.)
|
![( (](lp.gif) ![( (](lp.gif) ![W W](_cw.gif) ![{ {](lbrace.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![>. >.](rangle.gif) ![} }](rbrace.gif) ![: :](colon.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![} }](rbrace.gif) ![-1-1-onto-> -1-1-onto->](onetooneonto.gif) ![{ {](lbrace.gif) ![B B](_cb.gif) ![} }](rbrace.gif) ![) )](rp.gif) |
|
Theorem | f1sng 5417 |
A singleton of an ordered pair is a one-to-one function. (Contributed
by AV, 17-Apr-2021.)
|
![( (](lp.gif) ![( (](lp.gif) ![W W](_cw.gif) ![{ {](lbrace.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![>. >.](rangle.gif) ![} }](rbrace.gif) ![: :](colon.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![} }](rbrace.gif) ![-1-1-> -1-1->](onetoone.gif) ![W W](_cw.gif) ![) )](rp.gif) |
|
Theorem | fsnd 5418 |
A singleton of an ordered pair is a function. (Contributed by AV,
17-Apr-2021.)
|
![( (](lp.gif) ![V V](_cv.gif) ![( (](lp.gif) ![W W](_cw.gif) ![( (](lp.gif) ![{ {](lbrace.gif) ![<.
<.](langle.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![>. >.](rangle.gif) ![} }](rbrace.gif) ![: :](colon.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![} }](rbrace.gif) ![--> -->](longrightarrow.gif) ![W W](_cw.gif) ![) )](rp.gif) |
|
Theorem | f1oprg 5419 |
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.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![W W](_cw.gif) ![( (](lp.gif)
![Y Y](_cy.gif) ![) )](rp.gif)
![( (](lp.gif) ![( (](lp.gif) ![D D](_cd.gif) ![{ {](lbrace.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![>. >.](rangle.gif) ![<. <.](langle.gif) ![C C](_cc.gif) ![D D](_cd.gif) ![>. >.](rangle.gif) ![} }](rbrace.gif) ![: :](colon.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![C C](_cc.gif) ![} }](rbrace.gif) ![-1-1-onto-> -1-1-onto->](onetooneonto.gif) ![{ {](lbrace.gif) ![B B](_cb.gif)
![D D](_cd.gif) ![} }](rbrace.gif) ![)
)](rp.gif) ![) )](rp.gif) |
|
Theorem | tz6.12-2 5420* |
Function value when
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.)
|
![( (](lp.gif) ![E! E!](_e1.gif) ![A A](_ca.gif) ![F F](_cf.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![(/) (/)](varnothing.gif) ![) )](rp.gif) |
|
Theorem | fveu 5421* |
The value of a function at a unique point. (Contributed by Scott
Fenton, 6-Oct-2017.)
|
![( (](lp.gif) ![E! E!](_e1.gif) ![A A](_ca.gif) ![F F](_cf.gif)
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![U.
U.](bigcup.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![F F](_cf.gif) ![x x](_x.gif) ![} }](rbrace.gif) ![) )](rp.gif) |
|
Theorem | brprcneu 5422* |
If is a proper class
and is any class,
then there is no
unique set which is related to through the binary relation .
(Contributed by Scott Fenton, 7-Oct-2017.)
|
![( (](lp.gif) ![E! E!](_e1.gif) ![A A](_ca.gif) ![F F](_cf.gif) ![x x](_x.gif) ![) )](rp.gif) |
|
Theorem | fvprc 5423 |
A function's value at a proper class is the empty set. (Contributed by
NM, 20-May-1998.)
|
![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![(/) (/)](varnothing.gif) ![) )](rp.gif) |
|
Theorem | fv2 5424* |
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.)
|
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![U. U.](bigcup.gif) ![{ {](lbrace.gif) ![A. A.](forall.gif) ![y y](_y.gif) ![( (](lp.gif) ![A A](_ca.gif) ![F F](_cf.gif)
![x x](_x.gif) ![) )](rp.gif) ![} }](rbrace.gif) |
|
Theorem | dffv3g 5425* |
A definition of function value in terms of iota. (Contributed by Jim
Kingdon, 29-Dec-2018.)
|
![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![( (](lp.gif) ![iota iota](riota.gif) ![x x](_x.gif) ![( (](lp.gif) ![F F](_cf.gif) !["
"](backquote.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![} }](rbrace.gif) ![)
)](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | dffv4g 5426* |
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 4916), this definition
apparently does not appear in the literature. (Contributed by NM,
1-Aug-1994.)
|
![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![U. U.](bigcup.gif) ![{ {](lbrace.gif) ![( (](lp.gif) ![F F](_cf.gif) ![" "](backquote.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![} }](rbrace.gif) ![{ {](lbrace.gif) ![x x](_x.gif) ![} }](rbrace.gif) ![} }](rbrace.gif) ![) )](rp.gif) |
|
Theorem | elfv 5427* |
Membership in a function value. (Contributed by NM, 30-Apr-2004.)
|
![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![B B](_cb.gif) ![E. E.](exists.gif) ![x x](_x.gif) ![( (](lp.gif) ![A. A.](forall.gif) ![y y](_y.gif) ![( (](lp.gif) ![B B](_cb.gif) ![F F](_cf.gif) ![x x](_x.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fveq1 5428 |
Equality theorem for function value. (Contributed by NM,
29-Dec-1996.)
|
![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![( (](lp.gif) ![G G](_cg.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fveq2 5429 |
Equality theorem for function value. (Contributed by NM,
29-Dec-1996.)
|
![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fveq1i 5430 |
Equality inference for function value. (Contributed by NM,
2-Sep-2003.)
|
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![( (](lp.gif) ![G G](_cg.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | fveq1d 5431 |
Equality deduction for function value. (Contributed by NM,
2-Sep-2003.)
|
![( (](lp.gif) ![G G](_cg.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![( (](lp.gif) ![G G](_cg.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fveq2i 5432 |
Equality inference for function value. (Contributed by NM,
28-Jul-1999.)
|
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | fveq2d 5433 |
Equality deduction for function value. (Contributed by NM,
29-May-1999.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | 2fveq3 5434 |
Equality theorem for nested function values. (Contributed by AV,
14-Aug-2022.)
|
![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![( (](lp.gif) ![G G](_cg.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![) )](rp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![( (](lp.gif) ![G G](_cg.gif) ![` `](backtick.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fveq12i 5435 |
Equality deduction for function value. (Contributed by FL,
27-Jun-2014.)
|
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![( (](lp.gif) ![G G](_cg.gif) ![` `](backtick.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | fveq12d 5436 |
Equality deduction for function value. (Contributed by FL,
22-Dec-2008.)
|
![( (](lp.gif) ![G G](_cg.gif) ![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![( (](lp.gif) ![G G](_cg.gif) ![` `](backtick.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fveqeq2d 5437 |
Equality deduction for function value. (Contributed by BJ,
30-Aug-2022.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![( (](lp.gif) ![(
(](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![B B](_cb.gif)
![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fveqeq2 5438 |
Equality deduction for function value. (Contributed by BJ,
31-Aug-2022.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | nffv 5439 |
Bound-variable hypothesis builder for function value. (Contributed by
NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
|
![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | nffvmpt1 5440* |
Bound-variable hypothesis builder for mapping, special case.
(Contributed by Mario Carneiro, 25-Dec-2016.)
|
![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![` `](backtick.gif) ![C C](_cc.gif) ![) )](rp.gif) |
|
Theorem | nffvd 5441 |
Deduction version of bound-variable hypothesis builder nffv 5439.
(Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro,
15-Oct-2016.)
|
![( (](lp.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![F F](_cf.gif) ![( (](lp.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![A A](_ca.gif) ![( (](lp.gif) ![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | funfveu 5442* |
A function has one value given an argument in its domain. (Contributed
by Jim Kingdon, 29-Dec-2018.)
|
![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![E! E!](_e1.gif) ![A A](_ca.gif) ![F F](_cf.gif) ![y y](_y.gif) ![) )](rp.gif) |
|
Theorem | fvss 5443* |
The value of a function is a subset of if every element that could
be a candidate for the value is a subset of . (Contributed by
Mario Carneiro, 24-May-2019.)
|
![( (](lp.gif) ![A. A.](forall.gif) ![x x](_x.gif) ![( (](lp.gif) ![A A](_ca.gif) ![F F](_cf.gif) ![B B](_cb.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![) )](rp.gif) |
|
Theorem | fvssunirng 5444 |
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.)
|
![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![U. U.](bigcup.gif) ![F F](_cf.gif) ![) )](rp.gif) |
|
Theorem | relfvssunirn 5445 |
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.)
|
![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![U. U.](bigcup.gif) ![F F](_cf.gif) ![) )](rp.gif) |
|
Theorem | funfvex 5446 |
The value of a function exists. A special case of Corollary 6.13 of
[TakeutiZaring] p. 27.
(Contributed by Jim Kingdon, 29-Dec-2018.)
|
![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![_V _V](rmcv.gif) ![) )](rp.gif) |
|
Theorem | relrnfvex 5447 |
If a function has a set range, then the function value exists
unconditional on the domain. (Contributed by Mario Carneiro,
24-May-2019.)
|
![( (](lp.gif) ![( (](lp.gif) ![_V _V](rmcv.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![_V _V](rmcv.gif) ![) )](rp.gif) |
|
Theorem | fvexg 5448 |
Evaluating a set function at a set exists. (Contributed by Mario
Carneiro and Jim Kingdon, 28-May-2019.)
|
![( (](lp.gif) ![( (](lp.gif) ![W W](_cw.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![_V _V](rmcv.gif) ![) )](rp.gif) |
|
Theorem | fvex 5449 |
Evaluating a set function at a set exists. (Contributed by Mario
Carneiro and Jim Kingdon, 28-May-2019.)
|
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![_V _V](rmcv.gif) |
|
Theorem | sefvex 5450 |
If a function is set-like, then the function value exists if the input
does. (Contributed by Mario Carneiro, 24-May-2019.)
|
![( (](lp.gif) ![( (](lp.gif) ![`' `'](_cnv.gif) Se ![_V _V](rmcv.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![_V _V](rmcv.gif) ![) )](rp.gif) |
|
Theorem | fvifdc 5451 |
Move a conditional outside of a function. (Contributed by Jim Kingdon,
1-Jan-2022.)
|
DECID ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![if if](_if.gif) ![( (](lp.gif) ![ph ph](_varphi.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![) )](rp.gif) ![if if](_if.gif) ![( (](lp.gif) ![ph ph](_varphi.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![) )](rp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fv3 5452* |
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.)
|
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![{ {](lbrace.gif) ![( (](lp.gif) ![E. E.](exists.gif) ![y y](_y.gif) ![( (](lp.gif) ![A A](_ca.gif) ![F F](_cf.gif) ![y y](_y.gif) ![E! E!](_e1.gif)
![A A](_ca.gif) ![F F](_cf.gif) ![y y](_y.gif) ![) )](rp.gif) ![} }](rbrace.gif) |
|
Theorem | fvres 5453 |
The value of a restricted function. (Contributed by NM, 2-Aug-1994.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fvresd 5454 |
The value of a restricted function, deduction version of fvres 5453.
(Contributed by Glauco Siliprandi, 8-Apr-2021.)
|
![( (](lp.gif) ![B B](_cb.gif) ![( (](lp.gif) ![( (](lp.gif) ![(
(](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | funssfv 5455 |
The value of a member of the domain of a subclass of a function.
(Contributed by NM, 15-Aug-1994.)
|
![( (](lp.gif) ![( (](lp.gif)
![G G](_cg.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![( (](lp.gif) ![G G](_cg.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | tz6.12-1 5456* |
Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed
by NM, 30-Apr-2004.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![F F](_cf.gif) ![E! E!](_e1.gif) ![A A](_ca.gif) ![F F](_cf.gif) ![y y](_y.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![y y](_y.gif) ![) )](rp.gif) |
|
Theorem | tz6.12 5457* |
Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed
by NM, 10-Jul-1994.)
|
![( (](lp.gif) ![( (](lp.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![y y](_y.gif) ![E! E!](_e1.gif) ![y y](_y.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![y y](_y.gif) ![F F](_cf.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![y y](_y.gif) ![) )](rp.gif) |
|
Theorem | tz6.12f 5458* |
Function value, using bound-variable hypotheses instead of distinct
variable conditions. (Contributed by NM, 30-Aug-1999.)
|
![F/_ F/_](_finvbar.gif) ![y y](_y.gif) ![( (](lp.gif) ![( (](lp.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![y y](_y.gif) ![E! E!](_e1.gif) ![y y](_y.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![y y](_y.gif) ![F F](_cf.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![y y](_y.gif) ![) )](rp.gif) |
|
Theorem | tz6.12c 5459* |
Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by
NM, 30-Apr-2004.)
|
![( (](lp.gif) ![E! E!](_e1.gif) ![A A](_ca.gif) ![F F](_cf.gif) ![( (](lp.gif) ![(
(](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![A A](_ca.gif) ![F F](_cf.gif) ![y y](_y.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ndmfvg 5460 |
The value of a class outside its domain is the empty set. (Contributed
by Jim Kingdon, 15-Jan-2019.)
|
![( (](lp.gif) ![( (](lp.gif)
![F
F](_cf.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![(/) (/)](varnothing.gif) ![) )](rp.gif) |
|
Theorem | relelfvdm 5461 |
If a function value has a member, the argument belongs to the domain.
(Contributed by Jim Kingdon, 22-Jan-2019.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![B B](_cb.gif) ![) )](rp.gif) ![F F](_cf.gif) ![) )](rp.gif) |
|
Theorem | nfvres 5462 |
The value of a non-member of a restriction is the empty set.
(Contributed by NM, 13-Nov-1995.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![) )](rp.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![(/) (/)](varnothing.gif) ![) )](rp.gif) |
|
Theorem | nfunsn 5463 |
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.)
|
![( (](lp.gif) ![( (](lp.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![} }](rbrace.gif)
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![(/) (/)](varnothing.gif) ![) )](rp.gif) |
|
Theorem | 0fv 5464 |
Function value of the empty set. (Contributed by Stefan O'Rear,
26-Nov-2014.)
|
![( (](lp.gif) ![(/) (/)](varnothing.gif) ![`
`](backtick.gif) ![A A](_ca.gif) ![(/) (/)](varnothing.gif) |
|
Theorem | csbfv12g 5465 |
Move class substitution in and out of a function value. (Contributed by
NM, 11-Nov-2005.)
|
![( (](lp.gif) ![[_
[_](_ulbrack.gif) ![x x](_x.gif) ![]_ ]_](_urbrack.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![B B](_cb.gif)
![( (](lp.gif) ![[_ [_](_ulbrack.gif) ![x x](_x.gif) ![]_ ]_](_urbrack.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![[_
[_](_ulbrack.gif) ![x x](_x.gif) ![]_ ]_](_urbrack.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | csbfv2g 5466* |
Move class substitution in and out of a function value. (Contributed by
NM, 10-Nov-2005.)
|
![( (](lp.gif) ![[_
[_](_ulbrack.gif) ![x x](_x.gif) ![]_ ]_](_urbrack.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![B B](_cb.gif)
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![[_
[_](_ulbrack.gif) ![x x](_x.gif) ![]_ ]_](_urbrack.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | csbfvg 5467* |
Substitution for a function value. (Contributed by NM, 1-Jan-2006.)
|
![( (](lp.gif) ![[_
[_](_ulbrack.gif) ![x x](_x.gif) ![]_ ]_](_urbrack.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![x x](_x.gif)
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | funbrfv 5468 |
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.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![F F](_cf.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | funopfv 5469 |
The second element in an ordered pair member of a function is the
function's value. (Contributed by NM, 19-Jul-1996.)
|
![( (](lp.gif) ![( (](lp.gif) ![<.
<.](langle.gif) ![A A](_ca.gif) ![B B](_cb.gif)
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fnbrfvb 5470 |
Equivalence of function value and binary relation. (Contributed by NM,
19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![B B](_cb.gif) ![B B](_cb.gif) ![F F](_cf.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fnopfvb 5471 |
Equivalence of function value and ordered pair membership. (Contributed
by NM, 7-Nov-1995.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![B B](_cb.gif) ![<. <.](langle.gif) ![B B](_cb.gif) ![C C](_cc.gif) ![F F](_cf.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | funbrfvb 5472 |
Equivalence of function value and binary relation. (Contributed by NM,
26-Mar-2006.)
|
![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![A A](_ca.gif) ![F F](_cf.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | funopfvb 5473 |
Equivalence of function value and ordered pair membership. Theorem
4.3(ii) of [Monk1] p. 42. (Contributed by
NM, 26-Jan-1997.)
|
![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![<. <.](langle.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![F F](_cf.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | funbrfv2b 5474 |
Function value in terms of a binary relation. (Contributed by Mario
Carneiro, 19-Mar-2014.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![F F](_cf.gif) ![( (](lp.gif)
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | dffn5im 5475* |
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 5169 and dmmptss 5043. (Contributed by Jim Kingdon,
31-Dec-2018.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![x x](_x.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fnrnfv 5476* |
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.)
|
![( (](lp.gif) ![{ {](lbrace.gif) ![E. E.](exists.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![x x](_x.gif) ![) )](rp.gif) ![} }](rbrace.gif) ![) )](rp.gif) |
|
Theorem | fvelrnb 5477* |
A member of a function's range is a value of the function. (Contributed
by NM, 31-Oct-1995.)
|
![( (](lp.gif) ![( (](lp.gif) ![E. E.](exists.gif)
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![x x](_x.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | dfimafn 5478* |
Alternate definition of the image of a function. (Contributed by Raph
Levien, 20-Nov-2006.)
|
![( (](lp.gif) ![( (](lp.gif)
![F F](_cf.gif) ![( (](lp.gif) ![F F](_cf.gif) ![" "](backquote.gif) ![A A](_ca.gif)
![{ {](lbrace.gif) ![E. E.](exists.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![x x](_x.gif) ![y y](_y.gif) ![} }](rbrace.gif) ![) )](rp.gif) |
|
Theorem | dfimafn2 5479* |
Alternate definition of the image of a function as an indexed union of
singletons of function values. (Contributed by Raph Levien,
20-Nov-2006.)
|
![( (](lp.gif) ![( (](lp.gif)
![F F](_cf.gif) ![( (](lp.gif) ![F F](_cf.gif) ![" "](backquote.gif) ![A A](_ca.gif)
![U_ U_](_cupbar.gif) ![{ {](lbrace.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![x x](_x.gif) ![) )](rp.gif) ![} }](rbrace.gif) ![)
)](rp.gif) |
|
Theorem | funimass4 5480* |
Membership relation for the values of a function whose image is a
subclass. (Contributed by Raph Levien, 20-Nov-2006.)
|
![( (](lp.gif) ![( (](lp.gif)
![F F](_cf.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![" "](backquote.gif) ![A A](_ca.gif)
![A. A.](forall.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![x x](_x.gif) ![B B](_cb.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fvelima 5481* |
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.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) !["
"](backquote.gif) ![B B](_cb.gif) ![) )](rp.gif) ![E. E.](exists.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![x x](_x.gif) ![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | feqmptd 5482* |
Deduction form of dffn5im 5475. (Contributed by Mario Carneiro,
8-Jan-2015.)
|
![( (](lp.gif) ![F F](_cf.gif) ![: :](colon.gif) ![A A](_ca.gif) ![--> -->](longrightarrow.gif) ![B B](_cb.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![x x](_x.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | feqresmpt 5483* |
Express a restricted function as a mapping. (Contributed by Mario
Carneiro, 18-May-2016.)
|
![( (](lp.gif) ![F F](_cf.gif) ![: :](colon.gif) ![A A](_ca.gif) ![--> -->](longrightarrow.gif) ![B B](_cb.gif) ![( (](lp.gif) ![A A](_ca.gif) ![( (](lp.gif) ![( (](lp.gif) ![C C](_cc.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![x x](_x.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | dffn5imf 5484* |
Representation of a function in terms of its values. (Contributed by
Jim Kingdon, 31-Dec-2018.)
|
![F/_ F/_](_finvbar.gif) ![x x](_x.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![x x](_x.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fvelimab 5485* |
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.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![" "](backquote.gif) ![B B](_cb.gif) ![E. E.](exists.gif)
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![x x](_x.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fvi 5486 |
The value of the identity function. (Contributed by NM, 1-May-2004.)
(Revised by Mario Carneiro, 28-Apr-2015.)
|
![( (](lp.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![A A](_ca.gif) ![) )](rp.gif) |
|
Theorem | fniinfv 5487* |
The indexed intersection of a function's values is the intersection of
its range. (Contributed by NM, 20-Oct-2005.)
|
![( (](lp.gif) ![|^|_
|^|_](_capbar.gif)
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![x x](_x.gif) ![|^| |^|](bigcap.gif)
![F F](_cf.gif) ![) )](rp.gif) |
|
Theorem | fnsnfv 5488 |
Singleton of function value. (Contributed by NM, 22-May-1998.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![{ {](lbrace.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![B B](_cb.gif) ![) )](rp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![" "](backquote.gif) ![{ {](lbrace.gif) ![B B](_cb.gif) ![} }](rbrace.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fnimapr 5489 |
The image of a pair under a function. (Contributed by Jeff Madsen,
6-Jan-2011.)
|
![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![( (](lp.gif) ![F F](_cf.gif) ![" "](backquote.gif) ![{ {](lbrace.gif) ![B B](_cb.gif)
![C C](_cc.gif) ![} }](rbrace.gif) ![{ {](lbrace.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![B B](_cb.gif) ![) )](rp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![C C](_cc.gif) ![) )](rp.gif) ![} }](rbrace.gif) ![) )](rp.gif) |
|
Theorem | ssimaex 5490* |
The existence of a subimage. (Contributed by NM, 8-Apr-2007.)
|
![( (](lp.gif) ![( (](lp.gif)
![( (](lp.gif) ![F F](_cf.gif) ![" "](backquote.gif) ![A A](_ca.gif) ![) )](rp.gif) ![E. E.](exists.gif) ![x x](_x.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![" "](backquote.gif) ![x x](_x.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | ssimaexg 5491* |
The existence of a subimage. (Contributed by FL, 15-Apr-2007.)
|
![( (](lp.gif) ![( (](lp.gif)
![( (](lp.gif) ![F F](_cf.gif) ![" "](backquote.gif) ![A A](_ca.gif) ![) )](rp.gif) ![E. E.](exists.gif) ![x x](_x.gif) ![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![" "](backquote.gif) ![x x](_x.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | funfvdm 5492 |
A simplified expression for the value of a function when we know it's a
function. (Contributed by Jim Kingdon, 1-Jan-2019.)
|
![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![U. U.](bigcup.gif) ![( (](lp.gif) ![F F](_cf.gif) ![" "](backquote.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![} }](rbrace.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | funfvdm2 5493* |
The value of a function. Definition of function value in [Enderton]
p. 43. (Contributed by Jim Kingdon, 1-Jan-2019.)
|
![( (](lp.gif) ![( (](lp.gif) ![F F](_cf.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![U. U.](bigcup.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![F F](_cf.gif) ![y y](_y.gif) ![} }](rbrace.gif) ![) )](rp.gif) |
|
Theorem | funfvdm2f 5494 |
The value of a function. Version of funfvdm2 5493 using a bound-variable
hypotheses instead of distinct variable conditions. (Contributed by Jim
Kingdon, 1-Jan-2019.)
|
![F/_ F/_](_finvbar.gif) ![y y](_y.gif) ![F/_ F/_](_finvbar.gif) ![y y](_y.gif) ![( (](lp.gif) ![( (](lp.gif)
![F
F](_cf.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![U.
U.](bigcup.gif) ![{ {](lbrace.gif) ![A A](_ca.gif) ![F F](_cf.gif) ![y y](_y.gif) ![} }](rbrace.gif) ![)
)](rp.gif) |
|
Theorem | fvun1 5495 |
The value of a union when the argument is in the first domain.
(Contributed by Scott Fenton, 29-Jun-2013.)
|
![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![A A](_ca.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![G G](_cg.gif) ![) )](rp.gif) ![` `](backtick.gif) ![X X](_cx.gif)
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![X X](_cx.gif) ![) )](rp.gif) ![) )](rp.gif) |
|
Theorem | fvun2 5496 |
The value of a union when the argument is in the second domain.
(Contributed by Scott Fenton, 29-Jun-2013.)
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![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![( (](lp.gif) ![B B](_cb.gif) ![B B](_cb.gif) ![) )](rp.gif) ![( (](lp.gif) ![( (](lp.gif) ![G G](_cg.gif) ![) )](rp.gif) ![` `](backtick.gif) ![X X](_cx.gif)
![( (](lp.gif) ![G G](_cg.gif) ![` `](backtick.gif) ![X X](_cx.gif) ![) )](rp.gif) ![) )](rp.gif) |
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Theorem | dmfco 5497 |
Domains of a function composition. (Contributed by NM, 27-Jan-1997.)
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![( (](lp.gif) ![( (](lp.gif) ![G G](_cg.gif) ![( (](lp.gif) ![( (](lp.gif) ![G G](_cg.gif) ![( (](lp.gif) ![G G](_cg.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![F F](_cf.gif) ![) )](rp.gif) ![) )](rp.gif) |
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Theorem | fvco2 5498 |
Value of a function composition. Similar to second part of Theorem 3H
of [Enderton] p. 47. (Contributed by
NM, 9-Oct-2004.) (Proof shortened
by Andrew Salmon, 22-Oct-2011.) (Revised by Stefan O'Rear,
16-Oct-2014.)
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![( (](lp.gif) ![( (](lp.gif) ![A A](_ca.gif) ![( (](lp.gif) ![( (](lp.gif) ![G G](_cg.gif) ![) )](rp.gif) ![` `](backtick.gif) ![X X](_cx.gif)
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![( (](lp.gif) ![G G](_cg.gif) ![` `](backtick.gif) ![X X](_cx.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
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Theorem | fvco 5499 |
Value of a function composition. Similar to Exercise 5 of [TakeutiZaring]
p. 28. (Contributed by NM, 22-Apr-2006.) (Proof shortened by Mario
Carneiro, 26-Dec-2014.)
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![( (](lp.gif) ![( (](lp.gif) ![G G](_cg.gif) ![( (](lp.gif) ![( (](lp.gif) ![G G](_cg.gif) ![) )](rp.gif) ![` `](backtick.gif) ![A A](_ca.gif)
![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![( (](lp.gif) ![G G](_cg.gif) ![` `](backtick.gif) ![A A](_ca.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |
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Theorem | fvco3 5500 |
Value of a function composition. (Contributed by NM, 3-Jan-2004.)
(Revised by Mario Carneiro, 26-Dec-2014.)
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![( (](lp.gif) ![( (](lp.gif) ![G G](_cg.gif) ![: :](colon.gif) ![A A](_ca.gif) ![-->
-->](longrightarrow.gif) ![A A](_ca.gif)
![( (](lp.gif) ![( (](lp.gif) ![G G](_cg.gif) ![) )](rp.gif) ![` `](backtick.gif) ![C C](_cc.gif) ![( (](lp.gif) ![F F](_cf.gif) ![` `](backtick.gif) ![( (](lp.gif) ![G G](_cg.gif) ![` `](backtick.gif) ![C C](_cc.gif) ![) )](rp.gif) ![) )](rp.gif) ![) )](rp.gif) |