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Theorem List for Metamath Proof Explorer - 5401-5500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdff12 5401* Alternate definition of a one-to-one function. (Contributed by NM, 31-Dec-1996.)
 |-  ( F : A -1-1-> B  <-> 
 ( F : A --> B  /\  A. y E* x  x F y ) )
 
Theoremf1f 5402 A one-to-one mapping is a mapping. (Contributed by NM, 31-Dec-1996.)
 |-  ( F : A -1-1-> B 
 ->  F : A --> B )
 
Theoremf1fn 5403 A one-to-one mapping is a function on its domain. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-> B 
 ->  F  Fn  A )
 
Theoremf1fun 5404 A one-to-one mapping is a function. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-> B 
 ->  Fun  F )
 
Theoremf1rel 5405 A one-to-one onto mapping is a relation. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-> B 
 ->  Rel  F )
 
Theoremf1dm 5406 The domain of a one-to-one mapping. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-> B 
 ->  dom  F  =  A )
 
Theoremf1ss 5407 A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Mario Carneiro, 12-Jan-2013.)
 |-  ( ( F : A -1-1-> B  /\  B  C_  C )  ->  F : A -1-1-> C )
 
Theoremf1ssr 5408 Combine a one-to-one function with a restriction on the domain. (Contributed by Stefan O'Rear, 20-Feb-2015.)
 |-  ( ( F : A -1-1-> B  /\  ran  F  C_  C )  ->  F : A -1-1-> C )
 
Theoremf1ssres 5409 A function that is one-to-one is also one-to-one on some aubset of its domain. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-> B )
 
Theoremf1cnvcnv 5410 Two ways to express that a set  A (not necessarily a function) is one-to-one. Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one. We do not introduce a separate notation since we rarely use it. (Contributed by NM, 13-Aug-2004.)
 |-  ( `' `' A : dom  A -1-1-> _V  <->  ( Fun  `' A  /\  Fun  `' `' A ) )
 
Theoremf1co 5411 Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.)
 |-  ( ( F : B -1-1-> C  /\  G : A -1-1-> B )  ->  ( F  o.  G ) : A -1-1-> C )
 
Theoremfoeq1 5412 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( F  =  G  ->  ( F : A -onto-> B 
 <->  G : A -onto-> B ) )
 
Theoremfoeq2 5413 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F : A -onto-> C 
 <->  F : B -onto-> C ) )
 
Theoremfoeq3 5414 Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)
 |-  ( A  =  B  ->  ( F : C -onto-> A 
 <->  F : C -onto-> B ) )
 
Theoremnffo 5415 Bound-variable hypothesis builder for an onto function. (Contributed by NM, 16-May-2004.)
 |-  F/_ x F   &    |-  F/_ x A   &    |-  F/_ x B   =>    |- 
 F/ x  F : A -onto-> B
 
Theoremfof 5416 An onto mapping is a mapping. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A -onto-> B  ->  F : A --> B )
 
Theoremfofun 5417 An onto mapping is a function. (Contributed by NM, 29-Mar-2008.)
 |-  ( F : A -onto-> B  ->  Fun  F )
 
Theoremfofn 5418 An onto mapping is a function on its domain. (Contributed by NM, 16-Dec-2008.)
 |-  ( F : A -onto-> B  ->  F  Fn  A )
 
Theoremforn 5419 The codomain of an onto function is its range. (Contributed by NM, 3-Aug-1994.)
 |-  ( F : A -onto-> B  ->  ran  F  =  B )
 
Theoremdffo2 5420 Alternate definition of an onto function. (Contributed by NM, 22-Mar-2006.)
 |-  ( F : A -onto-> B 
 <->  ( F : A --> B  /\  ran  F  =  B ) )
 
Theoremfoima 5421 The image of the domain of an onto function. (Contributed by NM, 29-Nov-2002.)
 |-  ( F : A -onto-> B  ->  ( F " A )  =  B )
 
Theoremdffn4 5422 A function maps onto its range. (Contributed by NM, 10-May-1998.)
 |-  ( F  Fn  A  <->  F : A -onto-> ran  F )
 
Theoremfunforn 5423 A function maps its domain onto its range. (Contributed by NM, 23-Jul-2004.)
 |-  ( Fun  A  <->  A : dom  A -onto-> ran  A )
 
Theoremfodmrnu 5424 An onto function has unique domain and range. (Contributed by NM, 5-Nov-2006.)
 |-  ( ( F : A -onto-> B  /\  F : C -onto-> D )  ->  ( A  =  C  /\  B  =  D )
 )
 
Theoremfores 5425 Restriction of a function. (Contributed by NM, 4-Mar-1997.)
 |-  ( ( Fun  F  /\  A  C_  dom  F ) 
 ->  ( F  |`  A ) : A -onto-> ( F
 " A ) )
 
Theoremfoco 5426 Composition of onto functions. (Contributed by NM, 22-Mar-2006.)
 |-  ( ( F : B -onto-> C  /\  G : A -onto-> B )  ->  ( F  o.  G ) : A -onto-> C )
 
Theoremfoconst 5427 A nonzero constant function is onto. (Contributed by NM, 12-Jan-2007.)
 |-  ( ( F : A
 --> { B }  /\  F  =/=  (/) )  ->  F : A -onto-> { B } )
 
Theoremf1oeq1 5428 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
 |-  ( F  =  G  ->  ( F : A -1-1-onto-> B  <->  G : A -1-1-onto-> B ) )
 
Theoremf1oeq2 5429 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
 |-  ( A  =  B  ->  ( F : A -1-1-onto-> C  <->  F : B -1-1-onto-> C ) )
 
Theoremf1oeq3 5430 Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)
 |-  ( A  =  B  ->  ( F : C -1-1-onto-> A  <->  F : C -1-1-onto-> B ) )
 
Theoremf1oeq23 5431 Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  ( F : A
 -1-1-onto-> C 
 <->  F : B -1-1-onto-> D ) )
 
Theoremf1eq123d 5432 Equality deduction for one-to-one functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( F : A -1-1-> C  <->  G : B -1-1-> D ) )
 
Theoremfoeq123d 5433 Equality deduction for onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( F : A -onto-> C  <->  G : B -onto-> D ) )
 
Theoremf1oeq123d 5434 Equality deduction for one-to-one onto functions. (Contributed by Mario Carneiro, 27-Jan-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  ( F : A -1-1-onto-> C  <->  G : B -1-1-onto-> D ) )
 
Theoremnff1o 5435 Bound-variable hypothesis builder for a one-to-one onto function. (Contributed by NM, 16-May-2004.)
 |-  F/_ x F   &    |-  F/_ x A   &    |-  F/_ x B   =>    |- 
 F/ x  F : A
 -1-1-onto-> B
 
Theoremf1of1 5436 A one-to-one onto mapping is a one-to-one mapping. (Contributed by NM, 12-Dec-2003.)
 |-  ( F : A -1-1-onto-> B  ->  F : A -1-1-> B )
 
Theoremf1of 5437 A one-to-one onto mapping is a mapping. (Contributed by NM, 12-Dec-2003.)
 |-  ( F : A -1-1-onto-> B  ->  F : A --> B )
 
Theoremf1ofn 5438 A one-to-one onto mapping is function on its domain. (Contributed by NM, 12-Dec-2003.)
 |-  ( F : A -1-1-onto-> B  ->  F  Fn  A )
 
Theoremf1ofun 5439 A one-to-one onto mapping is a function. (Contributed by NM, 12-Dec-2003.)
 |-  ( F : A -1-1-onto-> B  ->  Fun  F )
 
Theoremf1orel 5440 A one-to-one onto mapping is a relation. (Contributed by NM, 13-Dec-2003.)
 |-  ( F : A -1-1-onto-> B  ->  Rel  F )
 
Theoremf1odm 5441 The domain of a one-to-one onto mapping. (Contributed by NM, 8-Mar-2014.)
 |-  ( F : A -1-1-onto-> B  ->  dom  F  =  A )
 
Theoremdff1o2 5442 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  Fun  `' F  /\  ran  F  =  B ) )
 
Theoremdff1o3 5443 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( F : A -1-1-onto-> B  <->  ( F : A -onto-> B  /\  Fun  `' F ) )
 
Theoremf1ofo 5444 A one-to-one onto function is an onto function. (Contributed by NM, 28-Apr-2004.)
 |-  ( F : A -1-1-onto-> B  ->  F : A -onto-> B )
 
Theoremdff1o4 5445 Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( F : A -1-1-onto-> B  <->  ( F  Fn  A  /\  `' F  Fn  B ) )
 
Theoremdff1o5 5446 Alternate definition of one-to-one onto function. (Contributed by NM, 10-Dec-2003.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( F : A -1-1-onto-> B  <->  ( F : A -1-1-> B  /\  ran  F  =  B ) )
 
Theoremf1orn 5447 A one-to-one function maps onto its range. (Contributed by NM, 13-Aug-2004.)
 |-  ( F : A -1-1-onto-> ran  F  <-> 
 ( F  Fn  A  /\  Fun  `' F ) )
 
Theoremf1f1orn 5448 A one-to-one function maps one-to-one onto its range. (Contributed by NM, 4-Sep-2004.)
 |-  ( F : A -1-1-> B 
 ->  F : A -1-1-onto-> ran  F )
 
Theoremf1oabexg 5449* The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  F  =  { f  |  ( f : A -1-1-onto-> B  /\  ph ) }   =>    |-  ( ( A  e.  C  /\  B  e.  D )  ->  F  e.  _V )
 
Theoremf1ocnv 5450 The converse of a one-to-one onto function is also one-to-one onto. (Contributed by NM, 11-Feb-1997.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
 |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
 
Theoremf1ocnvb 5451 A relation is a one-to-one onto function iff its converse is a one-to-one onto function with domain and range interchanged. (Contributed by NM, 8-Dec-2003.)
 |-  ( Rel  F  ->  ( F : A -1-1-onto-> B  <->  `' F : B -1-1-onto-> A ) )
 
Theoremf1ores 5452 The restriction of a one-to-one function maps one-to-one onto the image. (Contributed by NM, 25-Mar-1998.)
 |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( F  |`  C ) : C -1-1-onto-> ( F " C ) )
 
Theoremf1orescnv 5453 The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)
 |-  ( ( Fun  `' F  /\  ( F  |`  R ) : R -1-1-onto-> P )  ->  ( `' F  |`  P ) : P -1-1-onto-> R )
 
Theoremf1imacnv 5454 Preimage of an image. (Contributed by NM, 30-Sep-2004.)
 |-  ( ( F : A -1-1-> B  /\  C  C_  A )  ->  ( `' F " ( F
 " C ) )  =  C )
 
Theoremfoimacnv 5455 A reverse version of f1imacnv 5454. (Contributed by Jeffrey Hankins, 16-Jul-2009.)
 |-  ( ( F : A -onto-> B  /\  C  C_  B )  ->  ( F
 " ( `' F " C ) )  =  C )
 
Theoremfoun 5456 The union of two onto functions with disjoint domains is an onto function. (Contributed by Mario Carneiro, 22-Jun-2016.)
 |-  ( ( ( F : A -onto-> B  /\  G : C -onto-> D ) 
 /\  ( A  i^i  C )  =  (/) )  ->  ( F  u.  G ) : ( A  u.  C ) -onto-> ( B  u.  D ) )
 
Theoremf1oun 5457 The union of two one-to-one onto functions with disjoint domains and ranges. (Contributed by NM, 26-Mar-1998.)
 |-  ( ( ( F : A -1-1-onto-> B  /\  G : C
 -1-1-onto-> D )  /\  ( ( A  i^i  C )  =  (/)  /\  ( B  i^i  D )  =  (/) ) )  ->  ( F  u.  G ) : ( A  u.  C )
 -1-1-onto-> ( B  u.  D ) )
 
Theoremfun11iun 5458* 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  e.  _V   =>    |-  ( A. x  e.  A  ( B : D -1-1-> S  /\  A. y  e.  A  ( B  C_  C  \/  C  C_  B ) )  ->  U_ x  e.  A  B : U_ x  e.  A  D -1-1-> S )
 
Theoremresdif 5459 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 ) )
 
Theoremresin 5460 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 ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D ) 
 ->  ( F  |`  ( A  i^i  B ) ) : ( A  i^i  B ) -1-1-onto-> ( C  i^i  D ) )
 
Theoremf1oco 5461 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  o.  G ) : A -1-1-onto-> C )
 
Theoremf1cnv 5462 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 )
 
Theoremfuncocnv2 5463 Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( Fun  F  ->  ( F  o.  `' F )  =  (  _I  |` 
 ran  F ) )
 
Theoremfococnv2 5464 The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015.)
 |-  ( F : A -onto-> B  ->  ( F  o.  `' F )  =  (  _I  |`  B )
 )
 
Theoremf1ococnv2 5465 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  o.  `' F )  =  (  _I  |`  B ) )
 
Theoremf1cocnv2 5466 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
 |-  ( F : A -1-1-> B 
 ->  ( F  o.  `' F )  =  (  _I  |`  ran  F )
 )
 
Theoremf1ococnv1 5467 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  o.  F )  =  (  _I  |`  A ) )
 
Theoremf1cocnv1 5468 Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
 |-  ( F : A -1-1-> B 
 ->  ( `' F  o.  F )  =  (  _I  |`  A ) )
 
Theoremfuncoeqres 5469 Re-express a constraint on a composition as a constraint on the composand. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( ( Fun  G  /\  ( F  o.  G )  =  H )  ->  ( F  |`  ran  G )  =  ( H  o.  `' G ) )
 
Theoremffoss 5470* Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145. (Contributed by NM, 10-May-1998.)
 |-  F  e.  _V   =>    |-  ( F : A
 --> B  <->  E. x ( F : A -onto-> x  /\  x  C_  B ) )
 
Theoremf11o 5471* Relationship between one-to-one and one-to-one onto function. (Contributed by NM, 4-Apr-1998.)
 |-  F  e.  _V   =>    |-  ( F : A -1-1-> B  <->  E. x ( F : A -1-1-onto-> x  /\  x  C_  B ) )
 
Theoremf10 5472 The empty set maps one-to-one into any class. (Contributed by NM, 7-Apr-1998.)
 |-  (/) : (/) -1-1-> A
 
Theoremf1o00 5473 One-to-one onto mapping of the empty set. (Contributed by NM, 15-Apr-1998.)
 |-  ( F : (/) -1-1-onto-> A  <->  ( F  =  (/)  /\  A  =  (/) ) )
 
Theoremfo00 5474 Onto mapping of the empty set. (Contributed by NM, 22-Mar-2006.)
 |-  ( F : (/) -onto-> A  <-> 
 ( F  =  (/)  /\  A  =  (/) ) )
 
Theoremf1o0 5475 One-to-one onto mapping of the empty set. (Contributed by NM, 10-Sep-2004.)
 |-  (/) : (/)
 -1-1-onto-> (/)
 
Theoremf1oi 5476 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
 
Theoremf1ovi 5477 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
 
Theoremf1osn 5478 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  e.  _V   &    |-  B  e.  _V   =>    |- 
 { <. A ,  B >. } : { A }
 -1-1-onto-> { B }
 
Theoremf1osng 5479 A singleton of an ordered pair is one-to-one onto function. (Contributed by Mario Carneiro, 12-Jan-2013.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. } : { A } -1-1-onto-> { B } )
 
Theoremf1oprswap 5480 A two-element swap is a bijection on a pair. (Contributed by Mario Carneiro, 23-Jan-2015.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  { <. A ,  B >. ,  <. B ,  A >. } : { A ,  B } -1-1-onto-> { A ,  B }
 )
 
Theoremfv2 5481* 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 )  =  U. { x  |  A. y ( A F y  <->  y  =  x ) }
 
Theoremfvprc 5482 A function's value at a proper class is the empty set. (Contributed by NM, 20-May-1998.)
 |-  ( -.  A  e.  _V 
 ->  ( F `  A )  =  (/) )
 
Theoremelfv 5483* Membership in a function value. (Contributed by NM, 30-Apr-2004.)
 |-  ( A  e.  ( F `  B )  <->  E. x ( A  e.  x  /\  A. y ( B F y 
 <->  y  =  x ) ) )
 
Theoremfveq1 5484 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
 |-  ( F  =  G  ->  ( F `  A )  =  ( G `  A ) )
 
Theoremfveq2 5485 Equality theorem for function value. (Contributed by NM, 29-Dec-1996.)
 |-  ( A  =  B  ->  ( F `  A )  =  ( F `  B ) )
 
Theoremfveq1i 5486 Equality inference for function value. (Contributed by NM, 2-Sep-2003.)
 |-  F  =  G   =>    |-  ( F `  A )  =  ( G `  A )
 
Theoremfveq1d 5487 Equality deduction for function value. (Contributed by NM, 2-Sep-2003.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->  ( F `  A )  =  ( G `  A ) )
 
Theoremfveq2i 5488 Equality inference for function value. (Contributed by NM, 28-Jul-1999.)
 |-  A  =  B   =>    |-  ( F `  A )  =  ( F `  B )
 
Theoremfveq2d 5489 Equality deduction for function value. (Contributed by NM, 29-May-1999.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F `  A )  =  ( F `  B ) )
 
Theoremfveq12i 5490 Equality deduction for function value. (Contributed by FL, 27-Jun-2014.)
 |-  F  =  G   &    |-  A  =  B   =>    |-  ( F `  A )  =  ( G `  B )
 
Theoremfveq12d 5491 Equality deduction for function value. (Contributed by FL, 22-Dec-2008.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F `  A )  =  ( G `  B ) )
 
Theoremnffv 5492 Bound-variable hypothesis builder for function value. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  F/_ x F   &    |-  F/_ x A   =>    |-  F/_ x ( F `
  A )
 
Theoremnffvmpt1 5493* Bound-variable hypothesis builder for mapping, special case. (Contributed by Mario Carneiro, 25-Dec-2016.)
 |-  F/_ x ( ( x  e.  A  |->  B ) `
  C )
 
Theoremnffvd 5494 Deduction version of bound-variable hypothesis builder nffv 5492. (Contributed by NM, 10-Nov-2005.) (Revised by Mario Carneiro, 15-Oct-2016.)
 |-  ( ph  ->  F/_ x F )   &    |-  ( ph  ->  F/_ x A )   =>    |-  ( ph  ->  F/_ x ( F `  A ) )
 
Theoremcsbfv12g 5495 Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F `  B )  =  ( [_ A  /  x ]_ F ` 
 [_ A  /  x ]_ B ) )
 
Theoremcsbfv12gALT 5496 Move class substitution in and out of a function value.(This is csbfv12g 5495 with a shortened proof, shortened by Alan Sare, 10-Nov-2012.) The proof is derived from the virtual deduction proof csbfv12gALTVD 27943. Although the proof is shorter, the total number of steps of all theorems used in the proof is probably longer. (Contributed by NM, 10-Nov-2012.) (Proof modification is discouraged.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F `  B )  =  ( [_ A  /  x ]_ F ` 
 [_ A  /  x ]_ B ) )
 
Theoremcsbfv2g 5497* Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F `  B )  =  ( F ` 
 [_ A  /  x ]_ B ) )
 
Theoremcsbfvg 5498* Substitution for a function value. (Contributed by NM, 1-Jan-2006.)
 |-  ( A  e.  C  -> 
 [_ A  /  x ]_ ( F `  x )  =  ( F `  A ) )
 
Theoremfvex 5499 The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by NM, 30-Dec-1996.)
 |-  ( F `  A )  e.  _V
 
Theoremfvif 5500 Move a conditional outside of a function. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( F `  if ( ph ,  A ,  B ) )  =  if ( ph ,  ( F `  A ) ,  ( F `  B ) )
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