Theorem List for Intuitionistic Logic Explorer - 5301-5400 *Has distinct variable
group(s)
Type | Label | Description |
Statement |
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Theorem | fncnv 5301* |
Single-rootedness (see funcnv 5296) of a class cut down by a cross
product. (Contributed by NM, 5-Mar-2007.)
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Theorem | fun11 5302* |
Two ways of stating that is one-to-one (but not necessarily a
function). Each side is equivalent to Definition 6.4(3) of
[TakeutiZaring] p. 24, who use the
notation "Un2 (A)" for one-to-one
(but not necessarily a function). (Contributed by NM, 17-Jan-2006.)
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Theorem | fununi 5303* |
The union of a chain (with respect to inclusion) of functions is a
function. (Contributed by NM, 10-Aug-2004.)
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Theorem | funcnvuni 5304* |
The union of a chain (with respect to inclusion) of single-rooted sets
is single-rooted. (See funcnv 5296 for "single-rooted"
definition.)
(Contributed by NM, 11-Aug-2004.)
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Theorem | fun11uni 5305* |
The union of a chain (with respect to inclusion) of one-to-one functions
is a one-to-one function. (Contributed by NM, 11-Aug-2004.)
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Theorem | funin 5306 |
The intersection with a function is a function. Exercise 14(a) of
[Enderton] p. 53. (Contributed by NM,
19-Mar-2004.) (Proof shortened by
Andrew Salmon, 17-Sep-2011.)
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Theorem | funres11 5307 |
The restriction of a one-to-one function is one-to-one. (Contributed by
NM, 25-Mar-1998.)
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Theorem | funcnvres 5308 |
The converse of a restricted function. (Contributed by NM,
27-Mar-1998.)
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Theorem | cnvresid 5309 |
Converse of a restricted identity function. (Contributed by FL,
4-Mar-2007.)
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Theorem | funcnvres2 5310 |
The converse of a restriction of the converse of a function equals the
function restricted to the image of its converse. (Contributed by NM,
4-May-2005.)
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Theorem | funimacnv 5311 |
The image of the preimage of a function. (Contributed by NM,
25-May-2004.)
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Theorem | funimass1 5312 |
A kind of contraposition law that infers a subclass of an image from a
preimage subclass. (Contributed by NM, 25-May-2004.)
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Theorem | funimass2 5313 |
A kind of contraposition law that infers an image subclass from a subclass
of a preimage. (Contributed by NM, 25-May-2004.)
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Theorem | imadiflem 5314 |
One direction of imadif 5315. This direction does not require
 . (Contributed by Jim Kingdon,
25-Dec-2018.)
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Theorem | imadif 5315 |
The image of a difference is the difference of images. (Contributed by
NM, 24-May-1998.)
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Theorem | imainlem 5316 |
One direction of imain 5317. This direction does not require
 . (Contributed by Jim Kingdon,
25-Dec-2018.)
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Theorem | imain 5317 |
The image of an intersection is the intersection of images.
(Contributed by Paul Chapman, 11-Apr-2009.)
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Theorem | funimaexglem 5318 |
Lemma for funimaexg 5319. It constitutes the interesting part of
funimaexg 5319, in which
. (Contributed by Jim
Kingdon,
27-Dec-2018.)
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Theorem | funimaexg 5319 |
Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284.
Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM,
10-Sep-2006.)
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Theorem | funimaex 5320 |
The image of a set under any function is also a set. Equivalent of
Axiom of Replacement. Axiom 39(vi) of [Quine] p. 284. Compare Exercise
9 of [TakeutiZaring] p. 29.
(Contributed by NM, 17-Nov-2002.)
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Theorem | isarep1 5321* |
Part of a study of the Axiom of Replacement used by the Isabelle prover.
The object PrimReplace is apparently the image of the function encoded
by     i.e. the class          .
If so, we can prove Isabelle's "Axiom of Replacement"
conclusion without
using the Axiom of Replacement, for which I (N. Megill) currently have
no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by
Mario Carneiro, 4-Dec-2016.)
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 ![] ]](rbrack.gif)   |
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Theorem | isarep2 5322* |
Part of a study of the Axiom of Replacement used by the Isabelle prover.
In Isabelle, the sethood of PrimReplace is apparently postulated
implicitly by its type signature " i, i, i
=> o
=> i", which automatically asserts that it is a set without
using any
axioms. To prove that it is a set in Metamath, we need the hypotheses
of Isabelle's "Axiom of Replacement" as well as the Axiom of
Replacement
in the form funimaex 5320. (Contributed by NM, 26-Oct-2006.)
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         ![] ]](rbrack.gif)              |
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Theorem | fneq1 5323 |
Equality theorem for function predicate with domain. (Contributed by NM,
1-Aug-1994.)
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Theorem | fneq2 5324 |
Equality theorem for function predicate with domain. (Contributed by NM,
1-Aug-1994.)
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Theorem | fneq1d 5325 |
Equality deduction for function predicate with domain. (Contributed by
Paul Chapman, 22-Jun-2011.)
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Theorem | fneq2d 5326 |
Equality deduction for function predicate with domain. (Contributed by
Paul Chapman, 22-Jun-2011.)
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Theorem | fneq12d 5327 |
Equality deduction for function predicate with domain. (Contributed by
NM, 26-Jun-2011.)
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Theorem | fneq12 5328 |
Equality theorem for function predicate with domain. (Contributed by
Thierry Arnoux, 31-Jan-2017.)
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Theorem | fneq1i 5329 |
Equality inference for function predicate with domain. (Contributed by
Paul Chapman, 22-Jun-2011.)
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Theorem | fneq2i 5330 |
Equality inference for function predicate with domain. (Contributed by
NM, 4-Sep-2011.)
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Theorem | nffn 5331 |
Bound-variable hypothesis builder for a function with domain.
(Contributed by NM, 30-Jan-2004.)
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Theorem | fnfun 5332 |
A function with domain is a function. (Contributed by NM, 1-Aug-1994.)
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Theorem | fnrel 5333 |
A function with domain is a relation. (Contributed by NM, 1-Aug-1994.)
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Theorem | fndm 5334 |
The domain of a function. (Contributed by NM, 2-Aug-1994.)
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Theorem | funfni 5335 |
Inference to convert a function and domain antecedent. (Contributed by
NM, 22-Apr-2004.)
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Theorem | fndmu 5336 |
A function has a unique domain. (Contributed by NM, 11-Aug-1994.)
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Theorem | fnbr 5337 |
The first argument of binary relation on a function belongs to the
function's domain. (Contributed by NM, 7-May-2004.)
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Theorem | fnop 5338 |
The first argument of an ordered pair in a function belongs to the
function's domain. (Contributed by NM, 8-Aug-1994.)
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Theorem | fneu 5339* |
There is exactly one value of a function. (Contributed by NM,
22-Apr-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | fneu2 5340* |
There is exactly one value of a function. (Contributed by NM,
7-Nov-1995.)
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Theorem | fnun 5341 |
The union of two functions with disjoint domains. (Contributed by NM,
22-Sep-2004.)
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Theorem | fnunsn 5342 |
Extension of a function with a new ordered pair. (Contributed by NM,
28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
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Theorem | fnco 5343 |
Composition of two functions. (Contributed by NM, 22-May-2006.)
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Theorem | fnresdm 5344 |
A function does not change when restricted to its domain. (Contributed by
NM, 5-Sep-2004.)
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Theorem | fnresdisj 5345 |
A function restricted to a class disjoint with its domain is empty.
(Contributed by NM, 23-Sep-2004.)
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Theorem | 2elresin 5346 |
Membership in two functions restricted by each other's domain.
(Contributed by NM, 8-Aug-1994.)
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Theorem | fnssresb 5347 |
Restriction of a function with a subclass of its domain. (Contributed by
NM, 10-Oct-2007.)
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Theorem | fnssres 5348 |
Restriction of a function with a subclass of its domain. (Contributed by
NM, 2-Aug-1994.)
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Theorem | fnresin1 5349 |
Restriction of a function's domain with an intersection. (Contributed by
NM, 9-Aug-1994.)
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Theorem | fnresin2 5350 |
Restriction of a function's domain with an intersection. (Contributed by
NM, 9-Aug-1994.)
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Theorem | fnres 5351* |
An equivalence for functionality of a restriction. Compare dffun8 5263.
(Contributed by Mario Carneiro, 20-May-2015.)
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Theorem | fnresi 5352 |
Functionality and domain of restricted identity. (Contributed by NM,
27-Aug-2004.)
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Theorem | fnima 5353 |
The image of a function's domain is its range. (Contributed by NM,
4-Nov-2004.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | fn0 5354 |
A function with empty domain is empty. (Contributed by NM, 15-Apr-1998.)
(Proof shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | fnimadisj 5355 |
A class that is disjoint with the domain of a function has an empty image
under the function. (Contributed by FL, 24-Jan-2007.)
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Theorem | fnimaeq0 5356 |
Images under a function never map nonempty sets to empty sets.
(Contributed by Stefan O'Rear, 21-Jan-2015.)
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Theorem | dfmpt3 5357 |
Alternate definition for the maps-to notation df-mpt 4081. (Contributed
by Mario Carneiro, 30-Dec-2016.)
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Theorem | fnopabg 5358* |
Functionality and domain of an ordered-pair class abstraction.
(Contributed by NM, 30-Jan-2004.) (Proof shortened by Mario Carneiro,
4-Dec-2016.)
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Theorem | fnopab 5359* |
Functionality and domain of an ordered-pair class abstraction.
(Contributed by NM, 5-Mar-1996.)
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Theorem | mptfng 5360* |
The maps-to notation defines a function with domain. (Contributed by
Scott Fenton, 21-Mar-2011.)
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Theorem | fnmpt 5361* |
The maps-to notation defines a function with domain. (Contributed by
NM, 9-Apr-2013.)
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Theorem | mpt0 5362 |
A mapping operation with empty domain. (Contributed by Mario Carneiro,
28-Dec-2014.)
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Theorem | fnmpti 5363* |
Functionality and domain of an ordered-pair class abstraction.
(Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro,
31-Aug-2015.)
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Theorem | dmmpti 5364* |
Domain of an ordered-pair class abstraction that specifies a function.
(Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro,
31-Aug-2015.)
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Theorem | dmmptd 5365* |
The domain of the mapping operation, deduction form. (Contributed by
Glauco Siliprandi, 11-Dec-2019.)
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Theorem | mptun 5366 |
Union of mappings which are mutually compatible. (Contributed by Mario
Carneiro, 31-Aug-2015.)
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Theorem | feq1 5367 |
Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | feq2 5368 |
Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | feq3 5369 |
Equality theorem for functions. (Contributed by NM, 1-Aug-1994.)
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Theorem | feq23 5370 |
Equality theorem for functions. (Contributed by FL, 14-Jul-2007.) (Proof
shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | feq1d 5371 |
Equality deduction for functions. (Contributed by NM, 19-Feb-2008.)
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Theorem | feq2d 5372 |
Equality deduction for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | feq3d 5373 |
Equality deduction for functions. (Contributed by AV, 1-Jan-2020.)
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Theorem | feq12d 5374 |
Equality deduction for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | feq123d 5375 |
Equality deduction for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | feq123 5376 |
Equality theorem for functions. (Contributed by FL, 16-Nov-2008.)
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Theorem | feq1i 5377 |
Equality inference for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | feq2i 5378 |
Equality inference for functions. (Contributed by NM, 5-Sep-2011.)
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Theorem | feq23i 5379 |
Equality inference for functions. (Contributed by Paul Chapman,
22-Jun-2011.)
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Theorem | feq23d 5380 |
Equality deduction for functions. (Contributed by NM, 8-Jun-2013.)
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Theorem | nff 5381 |
Bound-variable hypothesis builder for a mapping. (Contributed by NM,
29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
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Theorem | sbcfng 5382* |
Distribute proper substitution through the function predicate with a
domain. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
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    ![]. ].](_drbrack.gif)   ![]_ ]_](_urbrack.gif)   ![]_ ]_](_urbrack.gif)    |
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Theorem | sbcfg 5383* |
Distribute proper substitution through the function predicate with
domain and codomain. (Contributed by Alexander van der Vekens,
15-Jul-2018.)
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    ![]. ].](_drbrack.gif)       ![]_ ]_](_urbrack.gif)     ![]_ ]_](_urbrack.gif)     ![]_ ]_](_urbrack.gif)    |
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Theorem | ffn 5384 |
A mapping is a function. (Contributed by NM, 2-Aug-1994.)
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Theorem | ffnd 5385 |
A mapping is a function with domain, deduction form. (Contributed by
Glauco Siliprandi, 17-Aug-2020.)
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Theorem | dffn2 5386 |
Any function is a mapping into . (Contributed by NM, 31-Oct-1995.)
(Proof shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | ffun 5387 |
A mapping is a function. (Contributed by NM, 3-Aug-1994.)
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Theorem | ffund 5388 |
A mapping is a function, deduction version. (Contributed by Glauco
Siliprandi, 3-Mar-2021.)
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Theorem | frel 5389 |
A mapping is a relation. (Contributed by NM, 3-Aug-1994.)
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Theorem | fdm 5390 |
The domain of a mapping. (Contributed by NM, 2-Aug-1994.)
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Theorem | fdmd 5391 |
Deduction form of fdm 5390. The domain of a mapping. (Contributed by
Glauco Siliprandi, 26-Jun-2021.)
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Theorem | fdmi 5392 |
The domain of a mapping. (Contributed by NM, 28-Jul-2008.)
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Theorem | frn 5393 |
The range of a mapping. (Contributed by NM, 3-Aug-1994.)
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Theorem | frnd 5394 |
Deduction form of frn 5393. The range of a mapping. (Contributed by
Glauco Siliprandi, 26-Jun-2021.)
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Theorem | dffn3 5395 |
A function maps to its range. (Contributed by NM, 1-Sep-1999.)
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Theorem | fss 5396 |
Expanding the codomain of a mapping. (Contributed by NM, 10-May-1998.)
(Proof shortened by Andrew Salmon, 17-Sep-2011.)
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Theorem | fssd 5397 |
Expanding the codomain of a mapping, deduction form. (Contributed by
Glauco Siliprandi, 11-Dec-2019.)
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Theorem | fssdmd 5398 |
Expressing that a class is a subclass of the domain of a function
expressed in maps-to notation, deduction form. (Contributed by AV,
21-Aug-2022.)
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Theorem | fssdm 5399 |
Expressing that a class is a subclass of the domain of a function
expressed in maps-to notation, semi-deduction form. (Contributed by AV,
21-Aug-2022.)
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Theorem | fco 5400 |
Composition of two mappings. (Contributed by NM, 29-Aug-1999.) (Proof
shortened by Andrew Salmon, 17-Sep-2011.)
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