MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ofc12 Structured version   Unicode version

Theorem ofc12 6564
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 465 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  W )
4 ofc12.3 . . . 4  |-  ( ph  ->  C  e.  X )
54adantr 465 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  X )
6 fconstmpt 5052 . . . 4  |-  ( A  X.  { B }
)  =  ( x  e.  A  |->  B )
76a1i 11 . . 3  |-  ( ph  ->  ( A  X.  { B } )  =  ( x  e.  A  |->  B ) )
8 fconstmpt 5052 . . . 4  |-  ( A  X.  { C }
)  =  ( x  e.  A  |->  C )
98a1i 11 . . 3  |-  ( ph  ->  ( A  X.  { C } )  =  ( x  e.  A  |->  C ) )
101, 3, 5, 7, 9offval2 6555 . 2  |-  ( ph  ->  ( ( A  X.  { B } )  oF R ( A  X.  { C }
) )  =  ( x  e.  A  |->  ( B R C ) ) )
11 fconstmpt 5052 . 2  |-  ( A  X.  { ( B R C ) } )  =  ( x  e.  A  |->  ( B R C ) )
1210, 11syl6eqr 2516 1  |-  ( ph  ->  ( ( A  X.  { B } )  oF R ( A  X.  { C }
) )  =  ( A  X.  { ( B R C ) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   {csn 4032    |-> cmpt 4515    X. cxp 5006  (class class class)co 6296    oFcof 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539
This theorem is referenced by:  pwsdiagmhm  16126  pwsdiaglmhm  17829  psrlmod  18180  coe1mul2  18436  itg2mulc  22279  dgrmulc  22793  mendlmod  31304  expgrowth  31402  lflvsdi2a  34906  lflvsass  34907  lflsc0N  34909
  Copyright terms: Public domain W3C validator