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Theorem ofc2 6331
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 5634 . . 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 3552 . 2  |-  ( A  i^i  A )  =  A
7 ofc2.4 . 2  |-  ( (
ph  /\  X  e.  A )  ->  ( F `  X )  =  C )
8 fvconst2g 5948 . . 3  |-  ( ( B  e.  W  /\  X  e.  A )  ->  ( ( A  X.  { B } ) `  X )  =  B )
92, 8sylan 459 . 2  |-  ( (
ph  /\  X  e.  A )  ->  (
( A  X.  { B } ) `  X
)  =  B )
101, 4, 5, 5, 6, 7, 9ofval 6317 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 360    = wceq 1653    e. wcel 1726   {csn 3816    X. cxp 4879    Fn wfn 5452   ` cfv 5457  (class class class)co 6084    o Fcof 6306
This theorem is referenced by:  lflvscl  29948  lkrsc  29968  ldualvsval  30009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308
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