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Theorem ofc2 6295
Description: Right operation by a constant. (Contributed by NM, 7-Oct-2014.)
Hypotheses
Ref Expression
ofc2.1  |-  ( ph  ->  A  e.  V )
ofc2.2  |-  ( ph  ->  B  e.  W )
ofc2.3  |-  ( ph  ->  F  Fn  A )
ofc2.4  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
Assertion
Ref Expression
ofc2  |-  ( (
ph  /\  X  e.  A )  ->  (
( F  o F R ( A  X.  { B } ) ) `
 X )  =  ( C R B ) )

Proof of Theorem ofc2
StepHypRef Expression
1 ofc2.3 . 2  |-  ( ph  ->  F  Fn  A )
2 ofc2.2 . . 3  |-  ( ph  ->  B  e.  W )
3 fnconstg 5598 . . 3  |-  ( B  e.  W  ->  ( A  X.  { B }
)  Fn  A )
42, 3syl 16 . 2  |-  ( ph  ->  ( A  X.  { B } )  Fn  A
)
5 ofc2.1 . 2  |-  ( ph  ->  A  e.  V )
6 inidm 3518 . 2  |-  ( A  i^i  A )  =  A
7 ofc2.4 . 2  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
8 fvconst2g 5912 . . 3  |-  ( ( B  e.  W  /\  X  e.  A )  ->  ( ( A  X.  { B } ) `  X )  =  B )
92, 8sylan 458 . 2  |-  ( (
ph  /\  X  e.  A )  ->  (
( A  X.  { B } ) `  X
)  =  B )
101, 4, 5, 5, 6, 7, 9ofval 6281 1  |-  ( (
ph  /\  X  e.  A )  ->  (
( F  o F R ( A  X.  { B } ) ) `
 X )  =  ( C R B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   {csn 3782    X. cxp 4843    Fn wfn 5416   ` cfv 5421  (class class class)co 6048    o Fcof 6270
This theorem is referenced by:  lflvscl  29572  lkrsc  29592  ldualvsval  29633
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pr 4371
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272
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