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Theorem caofid1 6543
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 5713 . . 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 5755 . . 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 5755 . . 3  |-  ( C  e.  X  ->  ( A  X.  { C }
)  Fn  A )
108, 9syl 16 . 2  |-  ( ph  ->  ( A  X.  { C } )  Fn  A
)
11 eqidd 2455 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
12 fvconst2g 6101 . . 3  |-  ( ( B  e.  W  /\  w  e.  A )  ->  ( ( A  X.  { B } ) `  w )  =  B )
135, 12sylan 469 . 2  |-  ( (
ph  /\  w  e.  A )  ->  (
( A  X.  { B } ) `  w
)  =  B )
142ffvelrnda 6007 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
15 caofid1.5 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  (
x R B )  =  C )
1615ralrimiva 2868 . . . . 5  |-  ( ph  ->  A. x  e.  S  ( x R B )  =  C )
17 oveq1 6277 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  (
x R B )  =  ( ( F `
 w ) R B ) )
1817eqeq1d 2456 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
( x R B )  =  C  <->  ( ( F `  w ) R B )  =  C ) )
1918rspccva 3206 . . . . 5  |-  ( ( A. x  e.  S  ( x R B )  =  C  /\  ( F `  w )  e.  S )  -> 
( ( F `  w ) R B )  =  C )
2016, 19sylan 469 . . . 4  |-  ( (
ph  /\  ( F `  w )  e.  S
)  ->  ( ( F `  w ) R B )  =  C )
2114, 20syldan 468 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R B )  =  C )
22 fvconst2g 6101 . . . 4  |-  ( ( C  e.  X  /\  w  e.  A )  ->  ( ( A  X.  { C } ) `  w )  =  C )
238, 22sylan 469 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  (
( A  X.  { C } ) `  w
)  =  C )
2421, 23eqtr4d 2498 . 2  |-  ( (
ph  /\  w  e.  A )  ->  (
( F `  w
) R B )  =  ( ( A  X.  { C }
) `  w )
)
251, 4, 7, 10, 11, 13, 24offveq 6534 1  |-  ( ph  ->  ( F  oF R ( A  X.  { B } ) )  =  ( A  X.  { C } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   {csn 4016    X. cxp 4986    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    oFcof 6511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513
This theorem is referenced by:  plymul0or  22843  fta1lem  22869  lfl0sc  35204
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