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Theorem caofcom 5786
Description: Transfer a commutative law to the function operation. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypotheses
Ref Expression
caofref.1  |-  ( ph  ->  A  e.  V )
caofref.2  |-  ( ph  ->  F : A --> S )
caofcom.3  |-  ( ph  ->  G : A --> S )
caofcom.4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x R y )  =  ( y R x ) )
Assertion
Ref Expression
caofcom  |-  ( ph  ->  ( F  oF R G )  =  ( G  oF R F ) )
Distinct variable groups:    x, y, F   
x, G, y    ph, x, y    x, R, y    x, S, y
Allowed substitution hints:    A( x, y)    V( x, y)

Proof of Theorem caofcom
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 caofref.2 . . . . . 6  |-  ( ph  ->  F : A --> S )
21ffvelrnda 5355 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
3 caofcom.3 . . . . . 6  |-  ( ph  ->  G : A --> S )
43ffvelrnda 5355 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
52, 4jca 300 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
)  e.  S  /\  ( G `  w )  e.  S ) )
6 caofcom.4 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x R y )  =  ( y R x ) )
76caovcomg 5708 . . . 4  |-  ( (
ph  /\  ( ( F `  w )  e.  S  /\  ( G `  w )  e.  S ) )  -> 
( ( F `  w ) R ( G `  w ) )  =  ( ( G `  w ) R ( F `  w ) ) )
85, 7syldan 276 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R ( G `
 w ) )  =  ( ( G `
 w ) R ( F `  w
) ) )
98mpteq2dva 3889 . 2  |-  ( ph  ->  ( w  e.  A  |->  ( ( F `  w ) R ( G `  w ) ) )  =  ( w  e.  A  |->  ( ( G `  w
) R ( F `
 w ) ) ) )
10 caofref.1 . . 3  |-  ( ph  ->  A  e.  V )
111feqmptd 5279 . . 3  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
123feqmptd 5279 . . 3  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
1310, 2, 4, 11, 12offval2 5778 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( w  e.  A  |->  ( ( F `  w ) R ( G `  w ) ) ) )
1410, 4, 2, 12, 11offval2 5778 . 2  |-  ( ph  ->  ( G  oF R F )  =  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) ) )
159, 13, 143eqtr4d 2125 1  |-  ( ph  ->  ( F  oF R G )  =  ( G  oF R F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434    |-> cmpt 3860   -->wf 4949   ` cfv 4953  (class class class)co 5564    oFcof 5762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-setind 4309
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-of 5764
This theorem is referenced by: (None)
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