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Theorem caofcom 6055
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 5602 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
3 caofcom.3 . . . . . 6  |-  ( ph  ->  G : A --> S )
43ffvelrnda 5602 . . . . 5  |-  ( (
ph  /\  w  e.  A )  ->  ( G `  w )  e.  S )
52, 4jca 304 . . . 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 5976 . . . 4  |-  ( (
ph  /\  ( ( F `  w )  e.  S  /\  ( G `  w )  e.  S ) )  -> 
( ( F `  w ) R ( G `  w ) )  =  ( ( G `  w ) R ( F `  w ) ) )
85, 7syldan 280 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R ( G `
 w ) )  =  ( ( G `
 w ) R ( F `  w
) ) )
98mpteq2dva 4054 . 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 5521 . . 3  |-  ( ph  ->  F  =  ( w  e.  A  |->  ( F `
 w ) ) )
123feqmptd 5521 . . 3  |-  ( ph  ->  G  =  ( w  e.  A  |->  ( G `
 w ) ) )
1310, 2, 4, 11, 12offval2 6047 . 2  |-  ( ph  ->  ( F  oF R G )  =  ( w  e.  A  |->  ( ( F `  w ) R ( G `  w ) ) ) )
1410, 4, 2, 12, 11offval2 6047 . 2  |-  ( ph  ->  ( G  oF R F )  =  ( w  e.  A  |->  ( ( G `  w ) R ( F `  w ) ) ) )
159, 13, 143eqtr4d 2200 1  |-  ( ph  ->  ( F  oF R G )  =  ( G  oF R F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128    |-> cmpt 4025   -->wf 5166   ` cfv 5170  (class class class)co 5824    oFcof 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-setind 4496
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-of 6032
This theorem is referenced by: (None)
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