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Theorem caofid1 6564
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  oF 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 5736 . . 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 5778 . . 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 5778 . . 3  |-  ( C  e.  X  ->  ( A  X.  { C }
)  Fn  A )
108, 9syl 16 . 2  |-  ( ph  ->  ( A  X.  { C } )  Fn  A
)
11 eqidd 2468 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
12 fvconst2g 6124 . . 3  |-  ( ( B  e.  W  /\  w  e.  A )  ->  ( ( A  X.  { B } ) `  w )  =  B )
135, 12sylan 471 . 2  |-  ( (
ph  /\  w  e.  A )  ->  (
( A  X.  { B } ) `  w
)  =  B )
142ffvelrnda 6031 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
15 caofid1.5 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  (
x R B )  =  C )
1615ralrimiva 2881 . . . . 5  |-  ( ph  ->  A. x  e.  S  ( x R B )  =  C )
17 oveq1 6301 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
x R B )  =  ( ( F `
 w ) R B ) )
1817eqeq1d 2469 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
( x R B )  =  C  <->  ( ( F `  w ) R B )  =  C ) )
1918rspccva 3218 . . . . 5  |-  ( ( A. x  e.  S  ( x R B )  =  C  /\  ( F `  w )  e.  S )  -> 
( ( F `  w ) R B )  =  C )
2016, 19sylan 471 . . . 4  |-  ( (
ph  /\  ( F `  w )  e.  S
)  ->  ( ( F `  w ) R B )  =  C )
2114, 20syldan 470 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R B )  =  C )
22 fvconst2g 6124 . . . 4  |-  ( ( C  e.  X  /\  w  e.  A )  ->  ( ( A  X.  { C } ) `  w )  =  C )
238, 22sylan 471 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( A  X.  { C } ) `  w
)  =  C )
2421, 23eqtr4d 2511 . 2  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R B )  =  ( ( A  X.  { C }
) `  w )
)
251, 4, 7, 10, 11, 13, 24offveq 6555 1  |-  ( ph  ->  ( F  oF R ( A  X.  { B } ) )  =  ( A  X.  { C } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   {csn 4032    X. cxp 5002    Fn wfn 5588   -->wf 5589   ` cfv 5593  (class class class)co 6294    oFcof 6532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-of 6534
This theorem is referenced by:  plymul0or  22521  fta1lem  22547  lfl0sc  34172
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