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Theorem ofc12 5762
Description: Function operation on two constant functions. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
ofc12.1  |-  ( ph  ->  A  e.  V )
ofc12.2  |-  ( ph  ->  B  e.  W )
ofc12.3  |-  ( ph  ->  C  e.  X )
Assertion
Ref Expression
ofc12  |-  ( ph  ->  ( ( A  X.  { B } )  oF R ( A  X.  { C }
) )  =  ( A  X.  { ( B R C ) } ) )

Proof of Theorem ofc12
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ofc12.1 . . 3  |-  ( ph  ->  A  e.  V )
2 ofc12.2 . . . 4  |-  ( ph  ->  B  e.  W )
32adantr 270 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  W )
4 ofc12.3 . . . 4  |-  ( ph  ->  C  e.  X )
54adantr 270 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  X )
6 fconstmpt 4413 . . . 4  |-  ( A  X.  { B }
)  =  ( x  e.  A  |->  B )
76a1i 9 . . 3  |-  ( ph  ->  ( A  X.  { B } )  =  ( x  e.  A  |->  B ) )
8 fconstmpt 4413 . . . 4  |-  ( A  X.  { C }
)  =  ( x  e.  A  |->  C )
98a1i 9 . . 3  |-  ( ph  ->  ( A  X.  { C } )  =  ( x  e.  A  |->  C ) )
101, 3, 5, 7, 9offval2 5757 . 2  |-  ( ph  ->  ( ( A  X.  { B } )  oF R ( A  X.  { C }
) )  =  ( x  e.  A  |->  ( B R C ) ) )
11 fconstmpt 4413 . 2  |-  ( A  X.  { ( B R C ) } )  =  ( x  e.  A  |->  ( B R C ) )
1210, 11syl6eqr 2132 1  |-  ( ph  ->  ( ( A  X.  { B } )  oF R ( A  X.  { C }
) )  =  ( A  X.  { ( B R C ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   {csn 3406    |-> cmpt 3847    X. cxp 4369  (class class class)co 5543    oFcof 5741
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 2064  ax-coll 3901  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-of 5743
This theorem is referenced by: (None)
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