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Theorem ofc12 5875
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 4485 . . . 4  |-  ( A  X.  { B }
)  =  ( x  e.  A  |->  B )
76a1i 9 . . 3  |-  ( ph  ->  ( A  X.  { B } )  =  ( x  e.  A  |->  B ) )
8 fconstmpt 4485 . . . 4  |-  ( A  X.  { C }
)  =  ( x  e.  A  |->  C )
98a1i 9 . . 3  |-  ( ph  ->  ( A  X.  { C } )  =  ( x  e.  A  |->  C ) )
101, 3, 5, 7, 9offval2 5870 . 2  |-  ( ph  ->  ( ( A  X.  { B } )  oF R ( A  X.  { C }
) )  =  ( x  e.  A  |->  ( B R C ) ) )
11 fconstmpt 4485 . 2  |-  ( A  X.  { ( B R C ) } )  =  ( x  e.  A  |->  ( B R C ) )
1210, 11syl6eqr 2138 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 1289    e. wcel 1438   {csn 3446    |-> cmpt 3899    X. cxp 4436  (class class class)co 5652    oFcof 5854
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-setind 4353
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-of 5856
This theorem is referenced by: (None)
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