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Theorem ofoprabco 25987
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 5848 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( F `  a )  e.  B )
4 ofoprabco.3 . . . . . . . 8  |-  ( ph  ->  G : A --> C )
54ffvelrnda 5848 . . . . . . 7  |-  ( (
ph  /\  a  e.  A )  ->  ( G `  a )  e.  C )
6 opelxpi 4876 . . . . . . 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 5788 . . . . 5  |-  ( (
ph  /\  a  e.  A )  ->  ( M `  a )  =  <. ( F `  a ) ,  ( G `  a )
>. )
98fveq2d 5700 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  ( N `  ( M `  a ) )  =  ( N `  <. ( F `  a ) ,  ( G `  a ) >. )
)
10 df-ov 6099 . . . . 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 6114 . . . . 5  |-  ( ( ( ph  /\  a  e.  A )  /\  (
x  =  ( F `
 a )  /\  y  =  ( G `  a ) ) )  ->  ( x R y )  =  ( ( F `  a
) R ( G `
 a ) ) )
17 ovex 6121 . . . . . 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 6223 . . . 4  |-  ( (
ph  /\  a  e.  A )  ->  (
( F `  a
) N ( G `
 a ) )  =  ( ( F `
 a ) R ( G `  a
) ) )
209, 11, 193eqtr2d 2481 . . 3  |-  ( (
ph  /\  a  e.  A )  ->  ( N `  ( M `  a ) )  =  ( ( F `  a ) R ( G `  a ) ) )
2120mpteq2dva 4383 . 2  |-  ( ph  ->  ( a  e.  A  |->  ( N `  ( M `  a )
) )  =  ( a  e.  A  |->  ( ( F `  a
) R ( G `
 a ) ) ) )
22 ovex 6121 . . . . . 6  |-  ( x R y )  e. 
_V
2322rgen2w 2789 . . . . 5  |-  A. x  e.  B  A. y  e.  C  ( x R y )  e. 
_V
24 eqid 2443 . . . . . 6  |-  ( x  e.  B ,  y  e.  C  |->  ( x R y ) )  =  ( x  e.  B ,  y  e.  C  |->  ( x R y ) )
2524fmpt2 6646 . . . . 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 5551 . . . 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 25978 . . 3  |-  ( ph  ->  M : A --> ( B  X.  C ) )
30 ofoprabco.1 . . . 4  |-  F/_ a M
3130fcomptf 25985 . . 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 5749 . . 3  |-  ( ph  ->  F  =  ( a  e.  A  |->  ( F `
 a ) ) )
354feqmptd 5749 . . 3  |-  ( ph  ->  G  =  ( a  e.  A  |->  ( G `
 a ) ) )
3633, 3, 5, 34, 35offval2 6341 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( a  e.  A  |->  ( ( F `  a ) R ( G `  a ) ) ) )
3721, 32, 363eqtr4rd 2486 1  |-  ( ph  ->  ( F  oF R G )  =  ( N  o.  M
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   F/_wnfc 2571   A.wral 2720   _Vcvv 2977   <.cop 3888    e. cmpt 4355    X. cxp 4843    o. ccom 4849   -->wf 5419   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098    oFcof 6323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-1st 6582  df-2nd 6583
This theorem is referenced by:  ofpreima  25989  rrvadd  26840
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