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Theorem ofoprabco 27319
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  oF 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 6032 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  a )  e.  B )
4 ofoprabco.3 . . . . . . . 8  |-  ( ph  ->  G : A --> C )
54ffvelrnda 6032 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( G `  a )  e.  C )
6 opelxpi 5037 . . . . . . 7  |-  ( ( ( F `  a
)  e.  B  /\  ( G `  a )  e.  C )  ->  <. ( F `  a
) ,  ( G `
 a ) >.  e.  ( B  X.  C
) )
73, 5, 6syl2anc 661 . . . . . 6  |-  ( (
ph  /\  a  e.  A )  ->  <. ( F `  a ) ,  ( G `  a ) >.  e.  ( B  X.  C ) )
81, 7fvmpt2d 5966 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( M `  a )  =  <. ( F `  a ) ,  ( G `  a )
>. )
98fveq2d 5876 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  ( N `  ( M `  a ) )  =  ( N `  <. ( F `  a ) ,  ( G `  a ) >. )
)
10 df-ov 6298 . . . . 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 465 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  N  =  ( x  e.  B ,  y  e.  C  |->  ( x R y ) ) )
14 simprl 755 . . . . . 6  |-  ( ( ( ph  /\  a  e.  A )  /\  (
x  =  ( F `
 a )  /\  y  =  ( G `  a ) ) )  ->  x  =  ( F `  a ) )
15 simprr 756 . . . . . 6  |-  ( ( ( ph  /\  a  e.  A )  /\  (
x  =  ( F `
 a )  /\  y  =  ( G `  a ) ) )  ->  y  =  ( G `  a ) )
1614, 15oveq12d 6313 . . . . 5  |-  ( ( ( ph  /\  a  e.  A )  /\  (
x  =  ( F `
 a )  /\  y  =  ( G `  a ) ) )  ->  ( x R y )  =  ( ( F `  a
) R ( G `
 a ) ) )
17 ovex 6320 . . . . . 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 6425 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  (
( F `  a
) N ( G `
 a ) )  =  ( ( F `
 a ) R ( G `  a
) ) )
209, 11, 193eqtr2d 2514 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  ( N `  ( M `  a ) )  =  ( ( F `  a ) R ( G `  a ) ) )
2120mpteq2dva 4539 . 2  |-  ( ph  ->  ( a  e.  A  |->  ( N `  ( M `  a )
) )  =  ( a  e.  A  |->  ( ( F `  a
) R ( G `
 a ) ) ) )
22 ovex 6320 . . . . . 6  |-  ( x R y )  e. 
_V
2322rgen2w 2829 . . . . 5  |-  A. x  e.  B  A. y  e.  C  ( x R y )  e. 
_V
24 eqid 2467 . . . . . 6  |-  ( x  e.  B ,  y  e.  C  |->  ( x R y ) )  =  ( x  e.  B ,  y  e.  C  |->  ( x R y ) )
2524fmpt2 6862 . . . . 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 208 . . . 4  |-  ( x  e.  B ,  y  e.  C  |->  ( x R y ) ) : ( B  X.  C ) --> _V
2712feq1d 5723 . . . 4  |-  ( ph  ->  ( N : ( B  X.  C ) --> _V  <->  ( x  e.  B ,  y  e.  C  |->  ( x R y ) ) : ( B  X.  C
) --> _V ) )
2826, 27mpbiri 233 . . 3  |-  ( ph  ->  N : ( B  X.  C ) --> _V )
291, 7fmpt3d 27310 . . 3  |-  ( ph  ->  M : A --> ( B  X.  C ) )
30 ofoprabco.1 . . . 4  |-  F/_ a M
3130fcomptf 27317 . . 3  |-  ( ( N : ( B  X.  C ) --> _V 
/\  M : A --> ( B  X.  C
) )  ->  ( N  o.  M )  =  ( a  e.  A  |->  ( N `  ( M `  a ) ) ) )
3228, 29, 31syl2anc 661 . 2  |-  ( ph  ->  ( N  o.  M
)  =  ( a  e.  A  |->  ( N `
 ( M `  a ) ) ) )
33 ofoprabco.4 . . 3  |-  ( ph  ->  A  e.  V )
342feqmptd 5927 . . 3  |-  ( ph  ->  F  =  ( a  e.  A  |->  ( F `
 a ) ) )
354feqmptd 5927 . . 3  |-  ( ph  ->  G  =  ( a  e.  A  |->  ( G `
 a ) ) )
3633, 3, 5, 34, 35offval2 6551 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( a  e.  A  |->  ( ( F `  a ) R ( G `  a ) ) ) )
3721, 32, 363eqtr4rd 2519 1  |-  ( ph  ->  ( F  oF R G )  =  ( N  o.  M
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   F/_wnfc 2615   A.wral 2817   _Vcvv 3118   <.cop 4039    |-> cmpt 4511    X. cxp 5003    o. ccom 5009   -->wf 5590   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297    oFcof 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-1st 6795  df-2nd 6796
This theorem is referenced by:  ofpreima  27321  rrvadd  28207
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