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Theorem ofoprabco 24032
Description: Function operation as a composition with an operation. (Contributed by Thierry Arnoux, 4-Jun-2017.)
Hypotheses
Ref Expression
ofoprabco.1  |-  F/_ a M
ofoprabco.2  |-  ( ph  ->  F : A --> B )
ofoprabco.3  |-  ( ph  ->  G : A --> C )
ofoprabco.4  |-  ( ph  ->  A  e.  V )
ofoprabco.5  |-  ( ph  ->  M  =  ( a  e.  A  |->  <. ( F `  a ) ,  ( G `  a ) >. )
)
ofoprabco.6  |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  C  |->  ( x R y ) ) )
Assertion
Ref Expression
ofoprabco  |-  ( ph  ->  ( F  o F R G )  =  ( N  o.  M
) )
Distinct variable groups:    x, a,
y, A    B, a, x, y    C, a, x, y    F, a, x, y    G, a, x, y    N, a    R, a, x, y    ph, a, x, y
Allowed substitution hints:    M( x, y, a)    N( x, y)    V( x, y, a)

Proof of Theorem ofoprabco
StepHypRef Expression
1 ofoprabco.5 . . . . . 6  |-  ( ph  ->  M  =  ( a  e.  A  |->  <. ( F `  a ) ,  ( G `  a ) >. )
)
2 ofoprabco.2 . . . . . . . 8  |-  ( ph  ->  F : A --> B )
32ffvelrnda 5829 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  a )  e.  B )
4 ofoprabco.3 . . . . . . . 8  |-  ( ph  ->  G : A --> C )
54ffvelrnda 5829 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( G `  a )  e.  C )
6 opelxpi 4869 . . . . . . 7  |-  ( ( ( F `  a
)  e.  B  /\  ( G `  a )  e.  C )  ->  <. ( F `  a
) ,  ( G `
 a ) >.  e.  ( B  X.  C
) )
73, 5, 6syl2anc 643 . . . . . 6  |-  ( (
ph  /\  a  e.  A )  ->  <. ( F `  a ) ,  ( G `  a ) >.  e.  ( B  X.  C ) )
81, 7fvmpt2d 5773 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( M `  a )  =  <. ( F `  a ) ,  ( G `  a )
>. )
98fveq2d 5691 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  ( N `  ( M `  a ) )  =  ( N `  <. ( F `  a ) ,  ( G `  a ) >. )
)
10 df-ov 6043 . . . . 5  |-  ( ( F `  a ) N ( G `  a ) )  =  ( N `  <. ( F `  a ) ,  ( G `  a ) >. )
1110a1i 11 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  (
( F `  a
) N ( G `
 a ) )  =  ( N `  <. ( F `  a
) ,  ( G `
 a ) >.
) )
12 ofoprabco.6 . . . . . 6  |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  C  |->  ( x R y ) ) )
1312adantr 452 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  N  =  ( x  e.  B ,  y  e.  C  |->  ( x R y ) ) )
14 simprl 733 . . . . . 6  |-  ( ( ( ph  /\  a  e.  A )  /\  (
x  =  ( F `
 a )  /\  y  =  ( G `  a ) ) )  ->  x  =  ( F `  a ) )
15 simprr 734 . . . . . 6  |-  ( ( ( ph  /\  a  e.  A )  /\  (
x  =  ( F `
 a )  /\  y  =  ( G `  a ) ) )  ->  y  =  ( G `  a ) )
1614, 15oveq12d 6058 . . . . 5  |-  ( ( ( ph  /\  a  e.  A )  /\  (
x  =  ( F `
 a )  /\  y  =  ( G `  a ) ) )  ->  ( x R y )  =  ( ( F `  a
) R ( G `
 a ) ) )
17 ovex 6065 . . . . . 6  |-  ( ( F `  a ) R ( G `  a ) )  e. 
_V
1817a1i 11 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  (
( F `  a
) R ( G `
 a ) )  e.  _V )
1913, 16, 3, 5, 18ovmpt2d 6160 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  (
( F `  a
) N ( G `
 a ) )  =  ( ( F `
 a ) R ( G `  a
) ) )
209, 11, 193eqtr2d 2442 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  ( N `  ( M `  a ) )  =  ( ( F `  a ) R ( G `  a ) ) )
2120mpteq2dva 4255 . 2  |-  ( ph  ->  ( a  e.  A  |->  ( N `  ( M `  a )
) )  =  ( a  e.  A  |->  ( ( F `  a
) R ( G `
 a ) ) ) )
22 ovex 6065 . . . . . 6  |-  ( x R y )  e. 
_V
2322rgen2w 2734 . . . . 5  |-  A. x  e.  B  A. y  e.  C  ( x R y )  e. 
_V
24 eqid 2404 . . . . . 6  |-  ( x  e.  B ,  y  e.  C  |->  ( x R y ) )  =  ( x  e.  B ,  y  e.  C  |->  ( x R y ) )
2524fmpt2 6377 . . . . 5  |-  ( A. x  e.  B  A. y  e.  C  (
x R y )  e.  _V  <->  ( x  e.  B ,  y  e.  C  |->  ( x R y ) ) : ( B  X.  C
) --> _V )
2623, 25mpbi 200 . . . 4  |-  ( x  e.  B ,  y  e.  C  |->  ( x R y ) ) : ( B  X.  C ) --> _V
2712feq1d 5539 . . . 4  |-  ( ph  ->  ( N : ( B  X.  C ) --> _V  <->  ( x  e.  B ,  y  e.  C  |->  ( x R y ) ) : ( B  X.  C
) --> _V ) )
2826, 27mpbiri 225 . . 3  |-  ( ph  ->  N : ( B  X.  C ) --> _V )
291, 7fmpt3d 24023 . . 3  |-  ( ph  ->  M : A --> ( B  X.  C ) )
30 ofoprabco.1 . . . 4  |-  F/_ a M
3130fcomptf 24030 . . 3  |-  ( ( N : ( B  X.  C ) --> _V 
/\  M : A --> ( B  X.  C
) )  ->  ( N  o.  M )  =  ( a  e.  A  |->  ( N `  ( M `  a ) ) ) )
3228, 29, 31syl2anc 643 . 2  |-  ( ph  ->  ( N  o.  M
)  =  ( a  e.  A  |->  ( N `
 ( M `  a ) ) ) )
33 ofoprabco.4 . . 3  |-  ( ph  ->  A  e.  V )
342feqmptd 5738 . . 3  |-  ( ph  ->  F  =  ( a  e.  A  |->  ( F `
 a ) ) )
354feqmptd 5738 . . 3  |-  ( ph  ->  G  =  ( a  e.  A  |->  ( G `
 a ) ) )
3633, 3, 5, 34, 35offval2 6281 . 2  |-  ( ph  ->  ( F  o F R G )  =  ( a  e.  A  |->  ( ( F `  a ) R ( G `  a ) ) ) )
3721, 32, 363eqtr4rd 2447 1  |-  ( ph  ->  ( F  o F R G )  =  ( N  o.  M
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   F/_wnfc 2527   A.wral 2666   _Vcvv 2916   <.cop 3777    e. cmpt 4226    X. cxp 4835    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042    o Fcof 6262
This theorem is referenced by:  ofpreima  24034  rrvadd  24663
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309
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