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Theorem ofoprabco 25917
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 5840 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  a )  e.  B )
4 ofoprabco.3 . . . . . . . 8  |-  ( ph  ->  G : A --> C )
54ffvelrnda 5840 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( G `  a )  e.  C )
6 opelxpi 4867 . . . . . . 7  |-  ( ( ( F `  a
)  e.  B  /\  ( G `  a )  e.  C )  ->  <. ( F `  a
) ,  ( G `
 a ) >.  e.  ( B  X.  C
) )
73, 5, 6syl2anc 656 . . . . . 6  |-  ( (
ph  /\  a  e.  A )  ->  <. ( F `  a ) ,  ( G `  a ) >.  e.  ( B  X.  C ) )
81, 7fvmpt2d 5780 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( M `  a )  =  <. ( F `  a ) ,  ( G `  a )
>. )
98fveq2d 5692 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  ( N `  ( M `  a ) )  =  ( N `  <. ( F `  a ) ,  ( G `  a ) >. )
)
10 df-ov 6093 . . . . 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 462 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  N  =  ( x  e.  B ,  y  e.  C  |->  ( x R y ) ) )
14 simprl 750 . . . . . 6  |-  ( ( ( ph  /\  a  e.  A )  /\  (
x  =  ( F `
 a )  /\  y  =  ( G `  a ) ) )  ->  x  =  ( F `  a ) )
15 simprr 751 . . . . . 6  |-  ( ( ( ph  /\  a  e.  A )  /\  (
x  =  ( F `
 a )  /\  y  =  ( G `  a ) ) )  ->  y  =  ( G `  a ) )
1614, 15oveq12d 6108 . . . . 5  |-  ( ( ( ph  /\  a  e.  A )  /\  (
x  =  ( F `
 a )  /\  y  =  ( G `  a ) ) )  ->  ( x R y )  =  ( ( F `  a
) R ( G `
 a ) ) )
17 ovex 6115 . . . . . 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 6217 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  (
( F `  a
) N ( G `
 a ) )  =  ( ( F `
 a ) R ( G `  a
) ) )
209, 11, 193eqtr2d 2479 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  ( N `  ( M `  a ) )  =  ( ( F `  a ) R ( G `  a ) ) )
2120mpteq2dva 4375 . 2  |-  ( ph  ->  ( a  e.  A  |->  ( N `  ( M `  a )
) )  =  ( a  e.  A  |->  ( ( F `  a
) R ( G `
 a ) ) ) )
22 ovex 6115 . . . . . 6  |-  ( x R y )  e. 
_V
2322rgen2w 2782 . . . . 5  |-  A. x  e.  B  A. y  e.  C  ( x R y )  e. 
_V
24 eqid 2441 . . . . . 6  |-  ( x  e.  B ,  y  e.  C  |->  ( x R y ) )  =  ( x  e.  B ,  y  e.  C  |->  ( x R y ) )
2524fmpt2 6640 . . . . 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 5543 . . . 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 25908 . . 3  |-  ( ph  ->  M : A --> ( B  X.  C ) )
30 ofoprabco.1 . . . 4  |-  F/_ a M
3130fcomptf 25915 . . 3  |-  ( ( N : ( B  X.  C ) --> _V 
/\  M : A --> ( B  X.  C
) )  ->  ( N  o.  M )  =  ( a  e.  A  |->  ( N `  ( M `  a ) ) ) )
3228, 29, 31syl2anc 656 . 2  |-  ( ph  ->  ( N  o.  M
)  =  ( a  e.  A  |->  ( N `
 ( M `  a ) ) ) )
33 ofoprabco.4 . . 3  |-  ( ph  ->  A  e.  V )
342feqmptd 5741 . . 3  |-  ( ph  ->  F  =  ( a  e.  A  |->  ( F `
 a ) ) )
354feqmptd 5741 . . 3  |-  ( ph  ->  G  =  ( a  e.  A  |->  ( G `
 a ) ) )
3633, 3, 5, 34, 35offval2 6335 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( a  e.  A  |->  ( ( F `  a ) R ( G `  a ) ) ) )
3721, 32, 363eqtr4rd 2484 1  |-  ( ph  ->  ( F  oF R G )  =  ( N  o.  M
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   F/_wnfc 2564   A.wral 2713   _Vcvv 2970   <.cop 3880    e. cmpt 4347    X. cxp 4834    o. ccom 4840   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092    oFcof 6317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-1st 6576  df-2nd 6577
This theorem is referenced by:  ofpreima  25919  rrvadd  26765
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