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Theorem ofc12 6321
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 4913 . . . 4  |-  ( A  X.  { B }
)  =  ( x  e.  A  |->  B )
76a1i 11 . . 3  |-  ( ph  ->  ( A  X.  { B } )  =  ( x  e.  A  |->  B ) )
8 fconstmpt 4913 . . . 4  |-  ( A  X.  { C }
)  =  ( x  e.  A  |->  C )
98a1i 11 . . 3  |-  ( ph  ->  ( A  X.  { C } )  =  ( x  e.  A  |->  C ) )
101, 3, 5, 7, 9offval2 6314 . 2  |-  ( ph  ->  ( ( A  X.  { B } )  o F R ( A  X.  { C }
) )  =  ( x  e.  A  |->  ( B R C ) ) )
11 fconstmpt 4913 . 2  |-  ( A  X.  { ( B R C ) } )  =  ( x  e.  A  |->  ( B R C ) )
1210, 11syl6eqr 2485 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 1652    e. wcel 1725   {csn 3806    e. cmpt 4258    X. cxp 4868  (class class class)co 6073    o Fcof 6295
This theorem is referenced by:  pwsdiagmhm  14760  pwsdiaglmhm  16125  psrlmod  16457  coe1mul2  16654  itg2mulc  19631  dgrmulc  20181  mendlmod  27469  expgrowth  27520  lflvsdi2a  29815  lflvsass  29816  lflsc0N  29818
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297
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