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Theorem ofc12 6268
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 } )  o F 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 452 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  W )
4 ofc12.3 . . . 4  |-  ( ph  ->  C  e.  X )
54adantr 452 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  X )
6 fconstmpt 4861 . . . 4  |-  ( A  X.  { B }
)  =  ( x  e.  A  |->  B )
76a1i 11 . . 3  |-  ( ph  ->  ( A  X.  { B } )  =  ( x  e.  A  |->  B ) )
8 fconstmpt 4861 . . . 4  |-  ( A  X.  { C }
)  =  ( x  e.  A  |->  C )
98a1i 11 . . 3  |-  ( ph  ->  ( A  X.  { C } )  =  ( x  e.  A  |->  C ) )
101, 3, 5, 7, 9offval2 6261 . 2  |-  ( ph  ->  ( ( A  X.  { B } )  o F R ( A  X.  { C }
) )  =  ( x  e.  A  |->  ( B R C ) ) )
11 fconstmpt 4861 . 2  |-  ( A  X.  { ( B R C ) } )  =  ( x  e.  A  |->  ( B R C ) )
1210, 11syl6eqr 2437 1  |-  ( ph  ->  ( ( A  X.  { B } )  o F R ( A  X.  { C }
) )  =  ( A  X.  { ( B R C ) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   {csn 3757    e. cmpt 4207    X. cxp 4816  (class class class)co 6020    o Fcof 6242
This theorem is referenced by:  pwsdiagmhm  14695  pwsdiaglmhm  16060  psrlmod  16392  coe1mul2  16589  itg2mulc  19506  dgrmulc  20056  mendlmod  27170  expgrowth  27221  lflvsdi2a  29195  lflvsass  29196  lflsc0N  29198
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244
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