P. 63 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 6201-6300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	caov31 6201*	Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( x F y ) = ( y F x )   &    |- ( ( x F y ) F z ) = ( x F ( y F z ) )   =>    |- ( ( A F B ) F C ) = ( ( C F B ) F A )
Theorem	caov13 6202*	Rearrange arguments in a commutative, associative operation. (Contributed by NM, 26-Aug-1995.)
|- A e. _V   &    |- B e. _V   &    |- C e. _V   &    |- ( x F y ) = ( y F x )   &    |- ( ( x F y ) F z ) = ( x F ( y F z ) )   =>    |- ( A F ( B F C ) ) = ( C F ( B F A ) )
Theorem	caovdilemd 6203*	Lemma used by real number construction. (Contributed by Jim Kingdon, 16-Sep-2019.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x G y ) = ( y G x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) G z ) = ( ( x G z ) F ( y G z ) ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x G y ) G z ) = ( x G ( y G z ) ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x G y ) e. S )   &    |- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   &    |- ( ph -> D e. S )   &    |- ( ph -> H e. S )   =>    |- ( ph -> ( ( ( A G C ) F ( B G D ) ) G H ) = ( ( A G ( C G H ) ) F ( B G ( D G H ) ) ) )
Theorem	caovlem2d 6204*	Rearrangement of expression involving multiplication ( G) and addition ( F). (Contributed by Jim Kingdon, 3-Jan-2020.)
|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x G y ) = ( y G x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) G z ) = ( ( x G z ) F ( y G z ) ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x G y ) G z ) = ( x G ( y G z ) ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x G y ) e. S )   &    |- ( ph -> A e. S )   &    |- ( ph -> B e. S )   &    |- ( ph -> C e. S )   &    |- ( ph -> D e. S )   &    |- ( ph -> H e. S )   &    |- ( ph -> R e. S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) = ( y F x ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x F y ) e. S )   =>    |- ( ph -> ( ( ( ( A G C ) F ( B G D ) ) G H ) F ( ( ( A G D ) F ( B G C ) ) G R ) ) = ( ( A G ( ( C G H ) F ( D G R ) ) ) F ( B G ( ( C G R ) F ( D G H ) ) ) ) )
Theorem	caovimo 6205*	Uniqueness of inverse element in commutative, associative operation with identity. The identity element is B. (Contributed by Jim Kingdon, 18-Sep-2019.)
|- B e. S   &    |- ( ( x e. S /\ y e. S ) -> ( x F y ) = ( y F x ) )   &    |- ( ( x e. S /\ y e. S /\ z e. S ) -> ( ( x F y ) F z ) = ( x F ( y F z ) ) )   &    |- ( x e. S -> ( x F B ) = x )   =>    |- ( A e. S -> E* w ( w e. S /\ ( A F w ) = B ) )
2.6.12  Maps-to notation
Theorem	elmpocl 6206*	If a two-parameter class is inhabited, constrain the implicit pair. (Contributed by Stefan O'Rear, 7-Mar-2015.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( X e. ( S F T ) -> ( S e. A /\ T e. B ) )
Theorem	elmpocl1 6207*	If a two-parameter class is inhabited, the first argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( X e. ( S F T ) -> S e. A )
Theorem	elmpocl2 6208*	If a two-parameter class is inhabited, the second argument is in its nominal domain. (Contributed by FL, 15-Oct-2012.) (Revised by Stefan O'Rear, 7-Mar-2015.)
|- F = ( x e. A , y e. B |-> C )   =>    |- ( X e. ( S F T ) -> T e. B )
Theorem	elovmpod 6209*	Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.) Variant of elovmpo 6210 in deduction form. (Revised by AV, 20-Apr-2025.)
|- O = ( a e. A , b e. B |-> C )   &    |- ( ph -> X e. A )   &    |- ( ph -> Y e. B )   &    |- ( ph -> D e. V )   &    |- ( ( a = X /\ b = Y ) -> C = D )   =>    |- ( ph -> ( E e. ( X O Y ) <-> E e. D ) )
Theorem	elovmpo 6210*	Utility lemma for two-parameter classes. (Contributed by Stefan O'Rear, 21-Jan-2015.)
|- D = ( a e. A , b e. B |-> C )   &    |- C e. _V   &    |- ( ( a = X /\ b = Y ) -> C = E )   =>    |- ( F e. ( X D Y ) <-> ( X e. A /\ Y e. B /\ F e. E ) )
Theorem	elovmporab 6211*	Implications for the value of an operation, defined by the maps-to notation with a class abstraction as a result, having an element. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
|- O = ( x e. _V , y e. _V |-> { z e. M | ph } )   &    |- ( ( X e. _V /\ Y e. _V ) -> M e. _V )   =>    |- ( Z e. ( X O Y ) -> ( X e. _V /\ Y e. _V /\ Z e. M ) )
Theorem	elovmporab1w 6212*	Implications for the value of an operation, defined by the maps-to notation with a class abstraction as a result, having an element. Here, the base set of the class abstraction depends on the first operand. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by GG, 26-Jan-2024.)
|- O = ( x e. _V , y e. _V |-> { z e. [_ x / m ]_ M | ph } )   &    |- ( ( X e. _V /\ Y e. _V ) -> [_ X / m ]_ M e. _V )   =>    |- ( Z e. ( X O Y ) -> ( X e. _V /\ Y e. _V /\ Z e. [_ X / m ]_ M ) )
Theorem	relmptopab 6213*	Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 7-Aug-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
|- F = ( x e. A |-> { <. y , z >. | ph } )   =>    |- Rel ( F ` B )
Theorem	f1ocnvd 6214*	Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
|- F = ( x e. A |-> C )   &    |- ( ( ph /\ x e. A ) -> C e. W )   &    |- ( ( ph /\ y e. B ) -> D e. X )   &    |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )   =>    |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )
Theorem	f1od 6215*	Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
|- F = ( x e. A |-> C )   &    |- ( ( ph /\ x e. A ) -> C e. W )   &    |- ( ( ph /\ y e. B ) -> D e. X )   &    |- ( ph -> ( ( x e. A /\ y = C ) <-> ( y e. B /\ x = D ) ) )   =>    |- ( ph -> F : A -1-1-onto-> B )
Theorem	f1ocnv2d 6216*	Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)
|- F = ( x e. A |-> C )   &    |- ( ( ph /\ x e. A ) -> C e. B )   &    |- ( ( ph /\ y e. B ) -> D e. A )   &    |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )   =>    |- ( ph -> ( F : A -1-1-onto-> B /\ `' F = ( y e. B |-> D ) ) )
Theorem	f1o2d 6217*	Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 12-May-2014.)
|- F = ( x e. A |-> C )   &    |- ( ( ph /\ x e. A ) -> C e. B )   &    |- ( ( ph /\ y e. B ) -> D e. A )   &    |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> ( x = D <-> y = C ) )   =>    |- ( ph -> F : A -1-1-onto-> B )
Theorem	f1opw2 6218*	A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6219 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.)
|- ( ph -> F : A -1-1-onto-> B )   &    |- ( ph -> ( `' F " a ) e. _V )   &    |- ( ph -> ( F " b ) e. _V )   =>    |- ( ph -> ( b e. ~P A |-> ( F " b ) ) : ~P A -1-1-onto-> ~P B )
Theorem	f1opw 6219*	A one-to-one mapping induces a one-to-one mapping on power sets. (Contributed by Stefan O'Rear, 18-Nov-2014.) (Revised by Mario Carneiro, 26-Jun-2015.)
|- ( F : A -1-1-onto-> B -> ( b e. ~P A |-> ( F " b ) ) : ~P A -1-1-onto-> ~P B )
Theorem	suppssfv 6220*	Formula building theorem for support restriction, on a function which preserves zero. (Contributed by Stefan O'Rear, 9-Mar-2015.)
|- ( ph -> ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) C_ L )   &    |- ( ph -> ( F ` Y ) = Z )   &    |- ( ( ph /\ x e. D ) -> A e. V )   =>    |- ( ph -> ( `' ( x e. D |-> ( F ` A ) ) " ( _V \ { Z } ) ) C_ L )
Theorem	suppssov1 6221*	Formula building theorem for support restrictions: operator with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
|- ( ph -> ( `' ( x e. D |-> A ) " ( _V \ { Y } ) ) C_ L )   &    |- ( ( ph /\ v e. R ) -> ( Y O v ) = Z )   &    |- ( ( ph /\ x e. D ) -> A e. V )   &    |- ( ( ph /\ x e. D ) -> B e. R )   =>    |- ( ph -> ( `' ( x e. D |-> ( A O B ) ) " ( _V \ { Z } ) ) C_ L )
2.6.13  Function operation
Syntax	cof 6222	Extend class notation to include mapping of an operation to a function operation.
class oF R
Syntax	cofr 6223	Extend class notation to include mapping of a binary relation to a function relation.
class oR R
Definition	df-of 6224*	Define the function operation map. The definition is designed so that if R is a binary operation, then oF R is the analogous operation on functions which corresponds to applying R pointwise to the values of the functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
|- oF R = ( f e. _V , g e. _V |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) )
Definition	df-ofr 6225*	Define the function relation map. The definition is designed so that if R is a binary relation, then oF R is the analogous relation on functions which is true when each element of the left function relates to the corresponding element of the right function. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- oR R = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) R ( g ` x ) }
Theorem	ofeqd 6226	Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.)
|- ( ph -> R = S )   =>    |- ( ph -> oF R = oF S )
Theorem	ofeq 6227	Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
|- ( R = S -> oF R = oF S )
Theorem	ofreq 6228	Equality theorem for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( R = S -> oR R = oR S )
Theorem	ofexg 6229	A function operation restricted to a set is a set. (Contributed by NM, 28-Jul-2014.)
|- ( A e. V -> ( oF R |` A ) e. _V )
Theorem	nfof 6230	Hypothesis builder for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
|- F/_ x R   =>    |- F/_ x oF R
Theorem	nfofr 6231	Hypothesis builder for function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- F/_ x R   =>    |- F/_ x oR R
Theorem	offval 6232*	Value of an operation applied to two functions. (Contributed by Mario Carneiro, 20-Jul-2014.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn B )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( A i^i B ) = S   &    |- ( ( ph /\ x e. A ) -> ( F ` x ) = C )   &    |- ( ( ph /\ x e. B ) -> ( G ` x ) = D )   =>    |- ( ph -> ( F oF R G ) = ( x e. S |-> ( C R D ) ) )
Theorem	ofrfval 6233*	Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn B )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( A i^i B ) = S   &    |- ( ( ph /\ x e. A ) -> ( F ` x ) = C )   &    |- ( ( ph /\ x e. B ) -> ( G ` x ) = D )   =>    |- ( ph -> ( F oR R G <-> A. x e. S C R D ) )
Theorem	ofvalg 6234	Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn B )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( A i^i B ) = S   &    |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )   &    |- ( ( ph /\ X e. B ) -> ( G ` X ) = D )   &    |- ( ( ph /\ X e. S ) -> ( C R D ) e. U )   =>    |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) )
Theorem	ofrval 6235	Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn B )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( A i^i B ) = S   &    |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )   &    |- ( ( ph /\ X e. B ) -> ( G ` X ) = D )   =>    |- ( ( ph /\ F oR R G /\ X e. S ) -> C R D )
Theorem	ofmresval 6236	Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
|- ( ph -> F e. A )   &    |- ( ph -> G e. B )   =>    |- ( ph -> ( F ( oF R |` ( A X. B ) ) G ) = ( F oF R G ) )
Theorem	off 6237*	The function operation produces a function. (Contributed by Mario Carneiro, 20-Jul-2014.)
|- ( ( ph /\ ( x e. S /\ y e. T ) ) -> ( x R y ) e. U )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> G : B --> T )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( A i^i B ) = C   =>    |- ( ph -> ( F oF R G ) : C --> U )
Theorem	offeq 6238*	Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Jim Kingdon, 26-Nov-2023.)
|- ( ( ph /\ ( x e. S /\ y e. T ) ) -> ( x R y ) e. U )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> G : B --> T )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( A i^i B ) = C   &    |- ( ph -> H : C --> U )   &    |- ( ( ph /\ x e. A ) -> ( F ` x ) = D )   &    |- ( ( ph /\ x e. B ) -> ( G ` x ) = E )   &    |- ( ( ph /\ x e. C ) -> ( D R E ) = ( H ` x ) )   =>    |- ( ph -> ( F oF R G ) = H )
Theorem	ofres 6239	Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn B )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( A i^i B ) = C   =>    |- ( ph -> ( F oF R G ) = ( ( F |` C ) oF R ( G |` C ) ) )
Theorem	offval2 6240*	The function operation expressed as a mapping. (Contributed by Mario Carneiro, 20-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ( ph /\ x e. A ) -> B e. W )   &    |- ( ( ph /\ x e. A ) -> C e. X )   &    |- ( ph -> F = ( x e. A |-> B ) )   &    |- ( ph -> G = ( x e. A |-> C ) )   =>    |- ( ph -> ( F oF R G ) = ( x e. A |-> ( B R C ) ) )
Theorem	ofrfval2 6241*	The function relation acting on maps. (Contributed by Mario Carneiro, 20-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ( ph /\ x e. A ) -> B e. W )   &    |- ( ( ph /\ x e. A ) -> C e. X )   &    |- ( ph -> F = ( x e. A |-> B ) )   &    |- ( ph -> G = ( x e. A |-> C ) )   =>    |- ( ph -> ( F oR R G <-> A. x e. A B R C ) )
Theorem	suppssof1 6242*	Formula building theorem for support restrictions: vector operation with left annihilator. (Contributed by Stefan O'Rear, 9-Mar-2015.)
|- ( ph -> ( `' A " ( _V \ { Y } ) ) C_ L )   &    |- ( ( ph /\ v e. R ) -> ( Y O v ) = Z )   &    |- ( ph -> A : D --> V )   &    |- ( ph -> B : D --> R )   &    |- ( ph -> D e. W )   =>    |- ( ph -> ( `' ( A oF O B ) " ( _V \ { Z } ) ) C_ L )
Theorem	ofco 6243	The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)
|- ( ph -> F Fn A )   &    |- ( ph -> G Fn B )   &    |- ( ph -> H : D --> C )   &    |- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> D e. X )   &    |- ( A i^i B ) = C   =>    |- ( ph -> ( ( F oF R G ) o. H ) = ( ( F o. H ) oF R ( G o. H ) ) )
Theorem	offveqb 6244*	Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
|- ( ph -> A e. V )   &    |- ( ph -> F Fn A )   &    |- ( ph -> G Fn A )   &    |- ( ph -> H Fn A )   &    |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )   &    |- ( ( ph /\ x e. A ) -> ( G ` x ) = C )   =>    |- ( ph -> ( H = ( F oF R G ) <-> A. x e. A ( H ` x ) = ( B R C ) ) )
Theorem	offveq 6245*	Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> F Fn A )   &    |- ( ph -> G Fn A )   &    |- ( ph -> H Fn A )   &    |- ( ( ph /\ x e. A ) -> ( F ` x ) = B )   &    |- ( ( ph /\ x e. A ) -> ( G ` x ) = C )   &    |- ( ( ph /\ x e. A ) -> ( B R C ) = ( H ` x ) )   =>    |- ( ph -> ( F oF R G ) = H )
Theorem	ofc1g 6246	Left operation by a constant. (Contributed by Mario Carneiro, 24-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> F Fn A )   &    |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )   &    |- ( ( ph /\ X e. A ) -> ( B R C ) e. U )   =>    |- ( ( ph /\ X e. A ) -> ( ( ( A X. { B } ) oF R F ) ` X ) = ( B R C ) )
Theorem	ofc2g 6247	Right operation by a constant. (Contributed by NM, 7-Oct-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> F Fn A )   &    |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )   &    |- ( ( ph /\ X e. A ) -> ( C R B ) e. U )   =>    |- ( ( ph /\ X e. A ) -> ( ( F oF R ( A X. { B } ) ) ` X ) = ( C R B ) )
Theorem	ofc12 6248	Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. X )   =>    |- ( ph -> ( ( A X. { B } ) oF R ( A X. { C } ) ) = ( A X. { ( B R C ) } ) )
Theorem	caofref 6249*	Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> F : A --> S )   &    |- ( ( ph /\ x e. S ) -> x R x )   =>    |- ( ph -> F oR R F )
Theorem	caofinvl 6250*	Transfer a left inverse law to the function operation. (Contributed by NM, 22-Oct-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> B e. W )   &    |- ( ph -> N : S --> S )   &    |- ( ph -> G = ( v e. A |-> ( N ` ( F ` v ) ) ) )   &    |- ( ( ph /\ x e. S ) -> ( ( N ` x ) R x ) = B )   =>    |- ( ph -> ( G oF R F ) = ( A X. { B } ) )
Theorem	caofid0l 6251*	Transfer a left identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> B e. W )   &    |- ( ( ph /\ x e. S ) -> ( B R x ) = x )   =>    |- ( ph -> ( ( A X. { B } ) oF R F ) = F )
Theorem	caofid0r 6252*	Transfer a right identity law to the function operation. (Contributed by NM, 21-Oct-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> B e. W )   &    |- ( ( ph /\ x e. S ) -> ( x R B ) = x )   =>    |- ( ph -> ( F oF R ( A X. { B } ) ) = F )
Theorem	caofid1 6253*	Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. X )   &    |- ( ( ph /\ x e. S ) -> ( x R B ) = C )   =>    |- ( ph -> ( F oF R ( A X. { B } ) ) = ( A X. { C } ) )
Theorem	caofid2 6254*	Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> B e. W )   &    |- ( ph -> C e. X )   &    |- ( ( ph /\ x e. S ) -> ( B R x ) = C )   =>    |- ( ph -> ( ( A X. { B } ) oF R F ) = ( A X. { C } ) )
Theorem	caofcom 6255*	Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> G : A --> S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x R y ) = ( y R x ) )   =>    |- ( ph -> ( F oF R G ) = ( G oF R F ) )
Theorem	caofrss 6256*	Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> G : A --> S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x R y -> x T y ) )   =>    |- ( ph -> ( F oR R G -> F oR T G ) )
Theorem	caoftrn 6257*	Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> F : A --> S )   &    |- ( ph -> G : A --> S )   &    |- ( ph -> H : A --> S )   &    |- ( ( ph /\ ( x e. S /\ y e. S /\ z e. S ) ) -> ( ( x R y /\ y T z ) -> x U z ) )   =>    |- ( ph -> ( ( F oR R G /\ G oR T H ) -> F oR U H ) )
Theorem	caofdig 6258*	Transfer a distributive law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> F : A --> K )   &    |- ( ph -> G : A --> S )   &    |- ( ph -> H : A --> S )   &    |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( x R y ) e. V )   &    |- ( ( ph /\ ( x e. K /\ y e. S ) ) -> ( x T y ) e. W )   &    |- ( ( ph /\ ( x e. K /\ y e. S /\ z e. S ) ) -> ( x T ( y R z ) ) = ( ( x T y ) O ( x T z ) ) )   =>    |- ( ph -> ( F oF T ( G oF R H ) ) = ( ( F oF T G ) oF O ( F oF T H ) ) )
2.6.14  Functions (continued)
Theorem	resfunexgALT 6259	The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5864 but requires ax-pow 4258 and ax-un 4524. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( Fun A /\ B e. C ) -> ( A |` B ) e. _V )
Theorem	cofunexg 6260	Existence of a composition when the first member is a function. (Contributed by NM, 8-Oct-2007.)
|- ( ( Fun A /\ B e. C ) -> ( A o. B ) e. _V )
Theorem	cofunex2g 6261	Existence of a composition when the second member is one-to-one. (Contributed by NM, 8-Oct-2007.)
|- ( ( A e. V /\ Fun `' B ) -> ( A o. B ) e. _V )
Theorem	fnexALT 6262	If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 5405. This version of fnex 5865 uses ax-pow 4258 and ax-un 4524, whereas fnex 5865 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( F Fn A /\ A e. B ) -> F e. _V )
Theorem	funexw 6263	Weak version of funex 5866 that holds without ax-coll 4199. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.)
|- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> F e. _V )
Theorem	mptexw 6264*	Weak version of mptex 5869 that holds without ax-coll 4199. If the domain and codomain of a function given by maps-to notation are sets, the function is a set. (Contributed by Rohan Ridenour, 13-Aug-2023.)
|- A e. _V   &    |- C e. _V   &    |- A. x e. A B e. C   =>    |- ( x e. A |-> B ) e. _V
Theorem	funrnex 6265	If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 5866. (Contributed by NM, 11-Nov-1995.)
|- ( dom F e. B -> ( Fun F -> ran F e. _V ) )
Theorem	focdmex 6266	If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
|- ( A e. C -> ( F : A -onto-> B -> B e. _V ) )
Theorem	f1dmex 6267	If the codomain of a one-to-one function exists, so does its domain. This can be thought of as a form of the Axiom of Replacement. (Contributed by NM, 4-Sep-2004.)
|- ( ( F : A -1-1-> B /\ B e. C ) -> A e. _V )
Theorem	abrexex 6268*	Existence of a class abstraction of existentially restricted sets. x is normally a free-variable parameter in the class expression substituted for B, which can be thought of as B ( x ). This simple-looking theorem is actually quite powerful and appears to involve the Axiom of Replacement in an intrinsic way, as can be seen by tracing back through the path mptexg 5868, funex 5866, fnex 5865, resfunexg 5864, and funimaexg 5405. See also abrexex2 6275. (Contributed by NM, 16-Oct-2003.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
|- A e. _V   =>    |- { y | E. x e. A y = B } e. _V
Theorem	abrexexg 6269*	Existence of a class abstraction of existentially restricted sets. x is normally a free-variable parameter in B. The antecedent assures us that A is a set. (Contributed by NM, 3-Nov-2003.)
|- ( A e. V -> { y | E. x e. A y = B } e. _V )
Theorem	iunexg 6270*	The existence of an indexed union. x is normally a free-variable parameter in B. (Contributed by NM, 23-Mar-2006.)
|- ( ( A e. V /\ A. x e. A B e. W ) -> U_ x e. A B e. _V )
Theorem	abrexex2g 6271*	Existence of an existentially restricted class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( A e. V /\ A. x e. A { y | ph } e. W ) -> { y | E. x e. A ph } e. _V )
Theorem	opabex3d 6272*	Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)
|- ( ph -> A e. _V )   &    |- ( ( ph /\ x e. A ) -> { y | ps } e. _V )   =>    |- ( ph -> { <. x , y >. | ( x e. A /\ ps ) } e. _V )
Theorem	opabex3 6273*	Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- A e. _V   &    |- ( x e. A -> { y | ph } e. _V )   =>    |- { <. x , y >. | ( x e. A /\ ph ) } e. _V
Theorem	iunex 6274*	The existence of an indexed union. x is normally a free-variable parameter in the class expression substituted for B, which can be read informally as B ( x ). (Contributed by NM, 13-Oct-2003.)
|- A e. _V   &    |- B e. _V   =>    |- U_ x e. A B e. _V
Theorem	abrexex2 6275*	Existence of an existentially restricted class abstraction. ph is normally has free-variable parameters x and y. See also abrexex 6268. (Contributed by NM, 12-Sep-2004.)
|- A e. _V   &    |- { y | ph } e. _V   =>    |- { y | E. x e. A ph } e. _V
Theorem	abexssex 6276*	Existence of a class abstraction with an existentially quantified expression. Both x and y can be free in ph. (Contributed by NM, 29-Jul-2006.)
|- A e. _V   &    |- { y | ph } e. _V   =>    |- { y | E. x ( x C_ A /\ ph ) } e. _V
Theorem	abexex 6277*	A condition where a class builder continues to exist after its wff is existentially quantified. (Contributed by NM, 4-Mar-2007.)
|- A e. _V   &    |- ( ph -> x e. A )   &    |- { y | ph } e. _V   =>    |- { y | E. x ph } e. _V
Theorem	oprabexd 6278*	Existence of an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ph -> A e. _V )   &    |- ( ph -> B e. _V )   &    |- ( ( ph /\ ( x e. A /\ y e. B ) ) -> E* z ps )   &    |- ( ph -> F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ps ) } )   =>    |- ( ph -> F e. _V )
Theorem	oprabex 6279*	Existence of an operation class abstraction. (Contributed by NM, 19-Oct-2004.)
|- A e. _V   &    |- B e. _V   &    |- ( ( x e. A /\ y e. B ) -> E* z ph )   &    |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ ph ) }   =>    |- F e. _V
Theorem	oprabex3 6280*	Existence of an operation class abstraction (special case). (Contributed by NM, 19-Oct-2004.)
|- H e. _V   &    |- F = { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\ y e. ( H X. H ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) }   =>    |- F e. _V
Theorem	oprabrexex2 6281*	Existence of an existentially restricted operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.)
|- A e. _V   &    |- { <. <. x , y >. , z >. | ph } e. _V   =>    |- { <. <. x , y >. , z >. | E. w e. A ph } e. _V
Theorem	ab2rexex 6282*	Existence of a class abstraction of existentially restricted sets. Variables x and y are normally free-variable parameters in the class expression substituted for C, which can be thought of as C ( x , y ). See comments for abrexex 6268. (Contributed by NM, 20-Sep-2011.)
|- A e. _V   &    |- B e. _V   =>    |- { z | E. x e. A E. y e. B z = C } e. _V
Theorem	ab2rexex2 6283*	Existence of an existentially restricted class abstraction. ph normally has free-variable parameters x, y, and z. Compare abrexex2 6275. (Contributed by NM, 20-Sep-2011.)
|- A e. _V   &    |- B e. _V   &    |- { z | ph } e. _V   =>    |- { z | E. x e. A E. y e. B ph } e. _V
Theorem	xpexgALT 6284	The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4833 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
|- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )
Theorem	offval3 6285*	General value of ( F oF R G ) with no assumptions on functionality of F and G. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- ( ( F e. V /\ G e. W ) -> ( F oF R G ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) )
Theorem	offres 6286	Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)
|- ( ( F e. V /\ G e. W ) -> ( ( F oF R G ) |` D ) = ( ( F |` D ) oF R ( G |` D ) ) )
Theorem	ofmres 6287*	Equivalent expressions for a restriction of the function operation map. Unlike oF R which is a proper class, ( oF R |` ( A X. B ) ) can be a set by ofmresex 6288, allowing it to be used as a function or structure argument. By ofmresval 6236, the restricted operation map values are the same as the original values, allowing theorems for oF R to be reused. (Contributed by NM, 20-Oct-2014.)
|- ( oF R |` ( A X. B ) ) = ( f e. A , g e. B |-> ( f oF R g ) )
Theorem	ofmresex 6288	Existence of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
|- ( ph -> A e. V )   &    |- ( ph -> B e. W )   =>    |- ( ph -> ( oF R |` ( A X. B ) ) e. _V )
Theorem	uchoice 6289*	Principle of unique choice. This is also called non-choice. The name choice results in its similarity to something like acfun 7400 (with the key difference being the change of E. to E!) but unique choice in fact follows from the axiom of collection and our other axioms. This is somewhat similar to Corollary 3.9.2 of [HoTT], p. (varies) but is better described by the paragraph at the end of Section 3.9 which starts "A similar issue arises in set-theoretic mathematics". (Contributed by Jim Kingdon, 13-Sep-2025.)
|- ( ( A e. V /\ A. x e. A E! y ph ) -> E. f ( f Fn A /\ A. x e. A [. ( f ` x ) / y ]. ph ) )
2.6.15  First and second members of an ordered pair
Syntax	c1st 6290	Extend the definition of a class to include the first member an ordered pair function.
class 1st
Syntax	c2nd 6291	Extend the definition of a class to include the second member an ordered pair function.
class 2nd
Definition	df-1st 6292	Define a function that extracts the first member, or abscissa, of an ordered pair. Theorem op1st 6298 proves that it does this. For example, ( 1st ` <. 3 , 4 >.) = 3 . Equivalent to Definition 5.13 (i) of [Monk1] p. 52 (compare op1sta 5210 and op1stb 4569). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
|- 1st = ( x e. _V |-> U. dom { x } )
Definition	df-2nd 6293	Define a function that extracts the second member, or ordinate, of an ordered pair. Theorem op2nd 6299 proves that it does this. For example, ( 2nd ` <. 3 , 4 >.) = 4 . Equivalent to Definition 5.13 (ii) of [Monk1] p. 52 (compare op2nda 5213 and op2ndb 5212). The notation is the same as Monk's. (Contributed by NM, 9-Oct-2004.)
|- 2nd = ( x e. _V |-> U. ran { x } )
Theorem	1stvalg 6294	The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( A e. _V -> ( 1st ` A ) = U. dom { A } )
Theorem	2ndvalg 6295	The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
|- ( A e. _V -> ( 2nd ` A ) = U. ran { A } )
Theorem	1st0 6296	The value of the first-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
|- ( 1st ` (/) ) = (/)
Theorem	2nd0 6297	The value of the second-member function at the empty set. (Contributed by NM, 23-Apr-2007.)
|- ( 2nd ` (/) ) = (/)
Theorem	op1st 6298	Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
|- A e. _V   &    |- B e. _V   =>    |- ( 1st ` <. A , B >. ) = A
Theorem	op2nd 6299	Extract the second member of an ordered pair. (Contributed by NM, 5-Oct-2004.)
|- A e. _V   &    |- B e. _V   =>    |- ( 2nd ` <. A , B >. ) = B
Theorem	op1std 6300	Extract the first member of an ordered pair. (Contributed by Mario Carneiro, 31-Aug-2015.)
|- A e. _V   &    |- B e. _V   =>    |- ( C = <. A , B >. -> ( 1st ` C ) = A )