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Theorem List for New Foundations Explorer - 5301-5400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremf1orescnv 5301 The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)
((Fun F (F R):R1-1-ontoP) → (F P):P1-1-ontoR)
 
Theoremf1imacnv 5302 Preimage of an image. (Contributed by set.mm contributors, 30-Sep-2004.)
((F:A1-1B C A) → (F “ (FC)) = C)
 
Theoremfoimacnv 5303 A reverse version of f1imacnv 5302. (Contributed by Jeffrey Hankins, 16-Jul-2009.)
((F:AontoB C B) → (F “ (FC)) = C)
 
Theoremf1oun 5304 The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by set.mm contributors, 26-Mar-1998.)
(((F:A1-1-ontoB G:C1-1-ontoD) ((AC) = (BD) = )) → (FG):(AC)–1-1-onto→(BD))
 
Theoremfun11iun 5305* 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 = yB = C)    &   B V       (x A (B:D1-1S y A (B C C B)) → x A B:x A D1-1S)
 
Theoremresdif 5306 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):AontoC (F B):BontoD) → (F (A B)):(A B)–1-1-onto→(C D))
 
Theoremresin 5307 The restriction of a one-to-one onto function to an intersection maps onto the intersection of the images. (Contributed by Paul Chapman, 11-Apr-2009.)
((Fun F (F A):AontoC (F B):BontoD) → (F (AB)):(AB)–1-1-onto→(CD))
 
Theoremf1oco 5308 Composition of one-to-one onto functions. (Contributed by set.mm contributors, 19-Mar-1998.)
((F:B1-1-ontoC G:A1-1-ontoB) → (F G):A1-1-ontoC)
 
Theoremf1ococnv2 5309 The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range. (Contributed by set.mm contributors, 13-Dec-2003.)
(F:A1-1-ontoB → (F F) = ( I B))
 
Theoremf1ococnv1 5310 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 set.mm contributors, 13-Dec-2003.)
(F:A1-1-ontoB → (F F) = ( I A))
 
Theoremf1cnv 5311 The converse of an injective function is bijective. (Contributed by FL, 11-Nov-2011.)
(F:A1-1BF:ran F1-1-ontoA)
 
Theoremf1cocnv1 5312 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
(F:A1-1B → (F F) = ( I A))
 
Theoremf1cocnv2 5313 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
(F:A1-1B → (F F) = ( I ran F))
 
Theoremffoss 5314* Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by set.mm contributors, 10-May-1998.)
F V       (F:A–→Bx(F:Aontox x B))
 
Theoremf11o 5315* Relationship between one-to-one and one-to-one onto function. (Contributed by set.mm contributors, 4-Apr-1998.)
F V       (F:A1-1Bx(F:A1-1-ontox x B))
 
Theoremf10 5316 The empty set maps one-to-one into any class. (Contributed by set.mm contributors, 7-Apr-1998.)
:1-1A
 
Theoremf1o00 5317 One-to-one onto mapping of the empty set. (Contributed by set.mm contributors, 15-Apr-1998.)
(F:1-1-ontoA ↔ (F = A = ))
 
Theoremfo00 5318 Onto mapping of the empty set. (Contributed by set.mm contributors, 22-Mar-2006.)
(F:ontoA ↔ (F = A = ))
 
Theoremf1o0 5319 One-to-one onto mapping of the empty set. (Contributed by set.mm contributors, 10-Feb-2004.) (Revised by set.mm contributors, 16-Feb-2004.)
:1-1-onto
 
Theoremf1oi 5320 A restriction of the identity relation is a one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 30-Apr-1998.) (Revised by set.mm contributors, 22-Oct-2011.)
( I A):A1-1-ontoA
 
Theoremf1ovi 5321 The identity relation is a one-to-one onto function on the universe. (Contributed by set.mm contributors, 16-May-2004.)
I :V–1-1-onto→V
 
Theoremf1osn 5322 A singleton of an ordered pair is one-to-one onto function. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 18-May-1998.) (Revised by set.mm contributors, 22-Oct-2011.)
A V    &   B V       {A, B}:{A}–1-1-onto→{B}
 
Theoremf1osng 5323 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})
 
Theoremfv2 5324* Alternate definition of function value. Definition 10.11 of [Quine] p. 68. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 30-Apr-2004.) (Revised by set.mm contributors, 18-Sep-2011.)
(FA) = {x y(AFyy = x)}
 
Theoremfvprc 5325 A function's value at a proper class is the empty set. (Contributed by set.mm contributors, 20-May-1998.)
A V → (FA) = )
 
Theoremelfv 5326* Membership in a function value. (Contributed by set.mm contributors, 30-Apr-2004.)
(A (FB) ↔ x(A x y(BFyy = x)))
 
Theoremfveq1 5327 Equality theorem for function value. (Contributed by set.mm contributors, 29-Dec-1996.)
(F = G → (FA) = (GA))
 
Theoremfveq2 5328 Equality theorem for function value. (Contributed by set.mm contributors, 29-Dec-1996.)
(A = B → (FA) = (FB))
 
Theoremfveq1i 5329 Equality inference for function value. (Contributed by set.mm contributors, 2-Sep-2003.)
F = G       (FA) = (GA)
 
Theoremfveq1d 5330 Equality deduction for function value. (Contributed by set.mm contributors, 2-Sep-2003.)
(φF = G)       (φ → (FA) = (GA))
 
Theoremfveq2i 5331 Equality inference for function value. (Contributed by set.mm contributors, 28-Jul-1999.)
A = B       (FA) = (FB)
 
Theoremfveq2d 5332 Equality deduction for function value. (Contributed by set.mm contributors, 29-May-1999.)
(φA = B)       (φ → (FA) = (FB))
 
Theoremfveq12d 5333 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
(φF = G)    &   (φA = B)       (φ → (FA) = (GB))
 
Theoremnffv 5334 Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
xF    &   xA       x(FA)
 
Theoremnffvd 5335 Deduction version of bound-variable hypothesis builder nffv 5334. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
(φxF)    &   (φxA)       (φx(FA))
 
Theoremcsbfv12g 5336 Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)
(A C[A / x](FB) = ([A / x]F[A / x]B))
 
Theoremcsbfv2g 5337* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
(A C[A / x](FB) = (F[A / x]B))
 
Theoremcsbfvg 5338* Substitution for a function value. (Contributed by NM, 1-Jan-2006.)
(A C[A / x](Fx) = (FA))
 
Theoremfvex 5339 The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by set.mm contributors, 30-Dec-1996.)
(FA) V
 
Theoremfvif 5340 Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
(F ‘ if(φ, A, B)) = if(φ, (FA), (FB))
 
Theoremfv3 5341* 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.)
(FA) = {x (y(x y AFy) ∃!y AFy)}
 
Theoremfvres 5342 The value of a restricted function. (Contributed by set.mm contributors, 2-Aug-1994.) (Revised by set.mm contributors, 16-Feb-2004.)
(A B → ((F B) ‘A) = (FA))
 
Theoremfunssfv 5343 The value of a member of the domain of a subclass of a function. (Contributed by set.mm contributors, 15-Aug-1994.) (Revised by set.mm contributors, 29-May-2007.)
((Fun F G F A dom G) → (FA) = (GA))
 
Theoremtz6.12-1 5344* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
((AFB ∃!y AFy) → (FA) = B)
 
Theoremtz6.12 5345* Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)
((A, B F ∃!yA, y F) → (FA) = B)
 
Theoremtz6.12-2 5346* Function value when F is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by set.mm contributors, 30-Apr-2004.)
∃!y AFy → (FA) = )
 
Theoremtz6.12c 5347* Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
(∃!y AFy → ((FA) = BAFB))
 
Theoremtz6.12i 5348 Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by set.mm contributors, 30-Apr-2004.) (Revised by set.mm contributors, 6-Apr-2007.)
(B → ((FA) = BAFB))
 
Theoremndmfv 5349 The value of a class outside its domain is the empty set. (Contributed by set.mm contributors, 24-Aug-1995.)
A dom F → (FA) = )
 
Theoremndmfvrcl 5350 Reverse closure law for function with the empty set not in its domain. (Contributed by set.mm contributors, 26-Apr-1996.)
dom F = S    &    ¬ S       ((FA) SA S)
 
Theoremelfvdm 5351 If a function value has a member, the argument belongs to the domain. (Contributed by set.mm contributors, 12-Feb-2007.)
(A (FB) → B dom F)
 
Theoremnfvres 5352 The value of a non-member of a restriction is the empty set. (Contributed by set.mm contributors, 13-Nov-1995.)
A B → ((F B) ‘A) = )
 
Theoremnfunsn 5353 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}) → (FA) = )
 
Theoremfv01 5354 Function value of the empty set. (Contributed by Stefan O'Rear, 26-Nov-2014.)
(A) =
 
Theoremfveqres 5355 Equal values imply equal values in a restriction. (Contributed by set.mm contributors, 13-Nov-1995.)
((FA) = (GA) → ((F B) ‘A) = ((G B) ‘A))
 
Theoremfunbrfv 5356 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 → (FA) = B))
 
Theoremfunopfv 5357 The second element in an ordered pair member of a function is the function's value. (Contributed by set.mm contributors, 19-Jul-1996.)
(Fun F → (A, B F → (FA) = B))
 
Theoremfnbrfvb 5358 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) → ((FB) = CBFC))
 
Theoremfnopfvb 5359 Equivalence of function value and ordered pair membership. (Contributed by set.mm contributors, 9-Jan-2015.)
((F Fn A B A) → ((FB) = CB, C F))
 
Theoremfunbrfvb 5360 Equivalence of function value and binary relation. (Contributed by set.mm contributors, 9-Jan-2015.)
((Fun F A dom F) → ((FA) = BAFB))
 
Theoremfunopfvb 5361 Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42. (Contributed by set.mm contributors, 9-Jan-2015.)
((Fun F A dom F) → ((FA) = BA, B F))
 
Theoremfunbrfv2b 5362 Function value in terms of a binary relation. (Contributed by Mario Carneiro, 19-Mar-2014.)
(Fun F → (AFB ↔ (A dom F (FA) = B)))
 
Theoremdffn5 5363* Representation of a function in terms of its values. (Contributed by set.mm contributors, 29-Jan-2004.)
(F Fn AF = {x, y (x A y = (Fx))})
 
Theoremfnrnfv 5364* The range of a function expressed as a collection of the function's values. (Contributed by set.mm contributors, 20-Oct-2005.)
(F Fn A → ran F = {y x A y = (Fx)})
 
Theoremfvelrnb 5365* A member of a function's range is a value of the function. (Contributed by set.mm contributors, 31-Oct-1995.)
(F Fn A → (B ran Fx A (Fx) = B))
 
Theoremdfimafn 5366* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)
((Fun F A dom F) → (FA) = {y x A (Fx) = y})
 
Theoremdfimafn2 5367* 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) → (FA) = x A {(Fx)})
 
Theoremfunimass4 5368* 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) → ((FA) Bx A (Fx) B))
 
Theoremfvelima 5369* Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 29-Apr-2004.) (Revised by set.mm contributors, 22-Oct-2011.)
((Fun F A (FB)) → x B (Fx) = A)
 
Theoremfvelimab 5370* Function value in an image. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (An unnecessary distinct variable restriction was removed by David Abernethy, 17-Dec-2011.) (Contributed by set.mm contributors, 20-Jan-2007.) (Revised by set.mm contributors, 25-Dec-2011.)
((F Fn A B A) → (C (FB) ↔ x B (Fx) = C))
 
Theoremfniniseg 5371 Membership in the preimage of a singleton, under a function. (Contributed by Mario Carneiro, 12-May-2014.)
(F Fn A → (C (F “ {B}) ↔ (C A (FC) = B)))
 
Theoremfniinfv 5372* The indexed intersection of a function's values is the intersection of its range. (Contributed by set.mm contributors, 20-Oct-2005.)
(F Fn Ax A (Fx) = ran F)
 
Theoremfnsnfv 5373 Singleton of function value. (Contributed by set.mm contributors, 22-May-1998.)
((F Fn A B A) → {(FB)} = (F “ {B}))
 
Theoremfnimapr 5374 The image of a pair under a funtion. (Contributed by Jeff Madsen, 6-Jan-2011.)
((F Fn A B A C A) → (F “ {B, C}) = {(FB), (FC)})
 
Theoremfunfv 5375 A simplified expression for the value of a function when we know it's a function. (Contributed by NM, 22-May-1998.)
(Fun F → (FA) = (F “ {A}))
 
Theoremfunfv2 5376* The value of a function. Definition of function value in [Enderton] p. 43. (Contributed by set.mm contributors, 22-May-1998.) (Revised by set.mm contributors, 11-May-2005.)
(Fun F → (FA) = {y AFy})
 
Theoremfunfv2f 5377 The value of a function. Version of funfv2 5376 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)
yA    &   yF       (Fun F → (FA) = {y AFy})
 
Theoremfvun 5378 Value of the union of two functions when the domains are separate. (Contributed by FL, 7-Nov-2011.)
(((Fun F Fun G) (dom F ∩ dom G) = ) → ((FG) ‘A) = ((FA) ∪ (GA)))
 
Theoremfvun1 5379 The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.)
((F Fn A G Fn B ((AB) = X A)) → ((FG) ‘X) = (FX))
 
Theoremfvun2 5380 The value of a union when the argument is in the second domain. (Contributed by Scott Fenton, 29-Jun-2013.)
((F Fn A G Fn B ((AB) = X B)) → ((FG) ‘X) = (GX))
 
Theoremdmfco 5381 Domains of a function composition. (Contributed by set.mm contributors, 27-Jan-1997.)
((Fun G A dom G) → (A dom (F G) ↔ (GA) dom F))
 
Theoremfvco2 5382 Value of a function composition. Similar to second part of Theorem 3H of [Enderton] p. 47. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 9-Oct-2004.) (Revised by set.mm contributors, 22-Oct-2011.)
((G Fn A C A) → ((F G) ‘C) = (F ‘(GC)))
 
Theoremfvco 5383 Value of a function composition. Similar to Exercise 5 of [TakeutiZaring] p. 28. (Contributed by set.mm contributors, 22-Apr-2006.)
((Fun G A dom G) → ((F G) ‘A) = (F ‘(GA)))
 
Theoremfvco3 5384 Value of a function composition. (Contributed by set.mm contributors, 3-Jan-2004.) (Revised by set.mm contributors, 21-Aug-2006.)
((G:A–→B C A) → ((F G) ‘C) = (F ‘(GC)))
 
Theoremfvopab4t 5385* Closed theorem form of fvopab4 5389. (Contributed by set.mm contributors, 21-Feb-2013.)
((xy(x = AB = C) x F = {x, y (x D y = B)} (A D C V)) → (FA) = C)
 
Theoremfvopab3g 5386* Value of a function given by ordered-pair class abstraction. (Contributed by set.mm contributors, 6-Mar-1996.)
B V    &   (x = A → (φψ))    &   (y = B → (ψχ))    &   (x C∃!yφ)    &   F = {x, y (x C φ)}       (A C → ((FA) = Bχ))
 
Theoremfvopab3ig 5387* Value of a function given by ordered-pair class abstraction. (Contributed by set.mm contributors, 23-Oct-1999.)
(x = A → (φψ))    &   (y = B → (ψχ))    &   (x C∃*yφ)    &   F = {x, y (x C φ)}       ((A C B D) → (χ → (FA) = B))
 
Theoremfvopab4g 5388* Value of a function given by ordered-pair class abstraction. (Contributed by set.mm contributors, 23-Oct-1999.)
(x = AB = C)    &   F = {x, y (x D y = B)}       ((A D C R) → (FA) = C)
 
Theoremfvopab4 5389* Value of a function given by ordered-pair class abstraction. (Contributed by set.mm contributors, 23-Oct-1999.)
(x = AB = C)    &   F = {x, y (x D y = B)}    &   C V       (A D → (FA) = C)
 
Theoremfvopab4ndm 5390* Value of a function given by an ordered-pair class abstraction, outside of its domain. (Contributed by set.mm contributors, 28-Mar-2008.)
F = {x, y (x A φ)}       B A → (FB) = )
 
Theoremfvopabg 5391* The value of a function given by ordered-pair class abstraction. (Contributed by set.mm contributors, 2-Sep-2003.)
(x = AB = C)       ((A V C W) → ({x, y y = B} ‘A) = C)
 
Theoremeqfnfv 5392* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 3-Aug-1994.) (Revised by set.mm contributors, 22-Oct-2011.)
((F Fn A G Fn A) → (F = Gx A (Fx) = (Gx)))
 
Theoremeqfnfv2 5393* Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by set.mm contributors, 3-Aug-1994.) (Revised by set.mm contributors, 5-Feb-2004.)
((F Fn A G Fn B) → (F = G ↔ (A = B x A (Fx) = (Gx))))
 
Theoremeqfnfv3 5394* Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009.)
((F Fn A G Fn B) → (F = G ↔ (B A x A (x B (Fx) = (Gx)))))
 
Theoremeqfnfvd 5395* Deduction for equality of functions. (Contributed by Mario Carneiro, 24-Jul-2014.)
(φF Fn A)    &   (φG Fn A)    &   ((φ x A) → (Fx) = (Gx))       (φF = G)
 
Theoremeqfnfv2f 5396* Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5392 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.)
xF    &   xG       ((F Fn A G Fn A) → (F = Gx A (Fx) = (Gx)))
 
Theoremeqfunfv 5397* Equality of functions is determined by their values. (Contributed by Scott Fenton, 19-Jun-2011.)
((Fun F Fun G) → (F = G ↔ (dom F = dom G x dom F(Fx) = (Gx))))
 
Theoremfvreseq 5398* Equality of restricted functions is determined by their values. (Contributed by set.mm contributors, 3-Aug-1994.) (Revised by set.mm contributors, 6-Feb-2004.)
(((F Fn A G Fn A) B A) → ((F B) = (G B) ↔ x B (Fx) = (Gx)))
 
Theoremchfnrn 5399* The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by set.mm contributors, 31-Aug-1999.)
((F Fn A x A (Fx) x) → ran F A)
 
Theoremfunfvop 5400 Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41. (Contributed by set.mm contributors, 14-Oct-1996.)
((Fun F A dom F) → A, (FA) F)
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