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Theorem caofid2 6556
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 )
caofid2.5  |-  ( (
ph  /\  x  e.  S )  ->  ( B R x )  =  C )
Assertion
Ref Expression
caofid2  |-  ( ph  ->  ( ( A  X.  { B } )  oF R F )  =  ( 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 caofid2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 caofref.1 . 2  |-  ( ph  ->  A  e.  V )
2 caofid0.3 . . 3  |-  ( ph  ->  B  e.  W )
3 fnconstg 5763 . . 3  |-  ( B  e.  W  ->  ( A  X.  { B }
)  Fn  A )
42, 3syl 16 . 2  |-  ( ph  ->  ( A  X.  { B } )  Fn  A
)
5 caofref.2 . . 3  |-  ( ph  ->  F : A --> S )
6 ffn 5721 . . 3  |-  ( F : A --> S  ->  F  Fn  A )
75, 6syl 16 . 2  |-  ( ph  ->  F  Fn  A )
8 caofid1.4 . . 3  |-  ( ph  ->  C  e.  X )
9 fnconstg 5763 . . 3  |-  ( C  e.  X  ->  ( A  X.  { C }
)  Fn  A )
108, 9syl 16 . 2  |-  ( ph  ->  ( A  X.  { C } )  Fn  A
)
11 fvconst2g 6109 . . 3  |-  ( ( B  e.  W  /\  w  e.  A )  ->  ( ( A  X.  { B } ) `  w )  =  B )
122, 11sylan 471 . 2  |-  ( (
ph  /\  w  e.  A )  ->  (
( A  X.  { B } ) `  w
)  =  B )
13 eqidd 2444 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
145ffvelrnda 6016 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
15 caofid2.5 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  ( B R x )  =  C )
1615ralrimiva 2857 . . . . 5  |-  ( ph  ->  A. x  e.  S  ( B R x )  =  C )
17 oveq2 6289 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  ( B R x )  =  ( B R ( F `  w ) ) )
1817eqeq1d 2445 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
( B R x )  =  C  <->  ( B R ( F `  w ) )  =  C ) )
1918rspccva 3195 . . . . 5  |-  ( ( A. x  e.  S  ( B R x )  =  C  /\  ( F `  w )  e.  S )  ->  ( B R ( F `  w ) )  =  C )
2016, 19sylan 471 . . . 4  |-  ( (
ph  /\  ( F `  w )  e.  S
)  ->  ( B R ( F `  w ) )  =  C )
2114, 20syldan 470 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( B R ( F `  w ) )  =  C )
22 fvconst2g 6109 . . . 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 2487 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( B R ( F `  w ) )  =  ( ( A  X.  { C } ) `  w ) )
251, 4, 7, 10, 12, 13, 24offveq 6546 1  |-  ( ph  ->  ( ( A  X.  { B } )  oF R F )  =  ( A  X.  { C } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   {csn 4014    X. cxp 4987    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281    oFcof 6523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525
This theorem is referenced by:  mbfmulc2lem  22032  i1fmulc  22088  itg1mulc  22089  itg2mulc  22132  dvcmulf  22326  coe0  22631  plymul0or  22655  expgrowth  31216
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