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Theorem caofcom 6125
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 )
21ffvelcdmda 5668 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
3 caofcom.3 . . . . . 6  |-  ( ph  ->  G : A --> S )
43ffvelcdmda 5668 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
52, 4jca 306 . . . 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 6048 . . . 4  |-  ( (
ph  /\  ( ( F `  w )  e.  S  /\  ( G `  w )  e.  S ) )  -> 
( ( F `  w ) R ( G `  w ) )  =  ( ( G `  w ) R ( F `  w ) ) )
85, 7syldan 282 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R ( G `
 w ) )  =  ( ( G `
 w ) R ( F `  w
) ) )
98mpteq2dva 4108 . 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 5586 . . 3  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
123feqmptd 5586 . . 3  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
1310, 2, 4, 11, 12offval2 6117 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( w  e.  A  |->  ( ( F `  w ) R ( G `  w ) ) ) )
1410, 4, 2, 12, 11offval2 6117 . 2  |-  ( ph  ->  ( G  oF R F )  =  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) ) )
159, 13, 143eqtr4d 2232 1  |-  ( ph  ->  ( F  oF R G )  =  ( G  oF R F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160    |-> cmpt 4079   -->wf 5228   ` cfv 5232  (class class class)co 5892    oFcof 6100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-setind 4551
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-ov 5895  df-oprab 5896  df-mpo 5897  df-of 6102
This theorem is referenced by: (None)
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