MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  caofid1 Unicode version

Theorem caofid1 6301
Description: Transfer a right absorption law to the function operation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
caofref.1  |-  ( ph  ->  A  e.  V )
caofref.2  |-  ( ph  ->  F : A --> S )
caofid0.3  |-  ( ph  ->  B  e.  W )
caofid1.4  |-  ( ph  ->  C  e.  X )
caofid1.5  |-  ( (
ph  /\  x  e.  S )  ->  (
x R B )  =  C )
Assertion
Ref Expression
caofid1  |-  ( ph  ->  ( F  o F R ( A  X.  { B } ) )  =  ( A  X.  { C } ) )
Distinct variable groups:    x, B    x, C    x, F    ph, x    x, R    x, S
Allowed substitution hints:    A( x)    V( x)    W( x)    X( x)

Proof of Theorem caofid1
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 caofref.1 . 2  |-  ( ph  ->  A  e.  V )
2 caofref.2 . . 3  |-  ( ph  ->  F : A --> S )
3 ffn 5558 . . 3  |-  ( F : A --> S  ->  F  Fn  A )
42, 3syl 16 . 2  |-  ( ph  ->  F  Fn  A )
5 caofid0.3 . . 3  |-  ( ph  ->  B  e.  W )
6 fnconstg 5598 . . 3  |-  ( B  e.  W  ->  ( A  X.  { B }
)  Fn  A )
75, 6syl 16 . 2  |-  ( ph  ->  ( A  X.  { B } )  Fn  A
)
8 caofid1.4 . . 3  |-  ( ph  ->  C  e.  X )
9 fnconstg 5598 . . 3  |-  ( C  e.  X  ->  ( A  X.  { C }
)  Fn  A )
108, 9syl 16 . 2  |-  ( ph  ->  ( A  X.  { C } )  Fn  A
)
11 eqidd 2413 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
12 fvconst2g 5912 . . 3  |-  ( ( B  e.  W  /\  w  e.  A )  ->  ( ( A  X.  { B } ) `  w )  =  B )
135, 12sylan 458 . 2  |-  ( (
ph  /\  w  e.  A )  ->  (
( A  X.  { B } ) `  w
)  =  B )
142ffvelrnda 5837 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
15 caofid1.5 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  (
x R B )  =  C )
1615ralrimiva 2757 . . . . 5  |-  ( ph  ->  A. x  e.  S  ( x R B )  =  C )
17 oveq1 6055 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
x R B )  =  ( ( F `
 w ) R B ) )
1817eqeq1d 2420 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
( x R B )  =  C  <->  ( ( F `  w ) R B )  =  C ) )
1918rspccva 3019 . . . . 5  |-  ( ( A. x  e.  S  ( x R B )  =  C  /\  ( F `  w )  e.  S )  -> 
( ( F `  w ) R B )  =  C )
2016, 19sylan 458 . . . 4  |-  ( (
ph  /\  ( F `  w )  e.  S
)  ->  ( ( F `  w ) R B )  =  C )
2114, 20syldan 457 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R B )  =  C )
22 fvconst2g 5912 . . . 4  |-  ( ( C  e.  X  /\  w  e.  A )  ->  ( ( A  X.  { C } ) `  w )  =  C )
238, 22sylan 458 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( A  X.  { C } ) `  w
)  =  C )
2421, 23eqtr4d 2447 . 2  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R B )  =  ( ( A  X.  { C }
) `  w )
)
251, 4, 7, 10, 11, 13, 24offveq 6292 1  |-  ( ph  ->  ( F  o F R ( A  X.  { B } ) )  =  ( A  X.  { C } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   {csn 3782    X. cxp 4843    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6048    o Fcof 6270
This theorem is referenced by:  plymul0or  20159  fta1lem  20185  lfl0sc  29577
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-13 1723  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-pow 4345  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
  Copyright terms: Public domain W3C validator