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Theorem caofid2 6327
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 } )  o F 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 5623 . . 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 5583 . . 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 5623 . . 3  |-  ( C  e.  X  ->  ( A  X.  { C }
)  Fn  A )
108, 9syl 16 . 2  |-  ( ph  ->  ( A  X.  { C } )  Fn  A
)
11 fvconst2g 5937 . . 3  |-  ( ( B  e.  W  /\  w  e.  A )  ->  ( ( A  X.  { B } ) `  w )  =  B )
122, 11sylan 458 . 2  |-  ( (
ph  /\  w  e.  A )  ->  (
( A  X.  { B } ) `  w
)  =  B )
13 eqidd 2436 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
145ffvelrnda 5862 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
15 caofid2.5 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  ( B R x )  =  C )
1615ralrimiva 2781 . . . . 5  |-  ( ph  ->  A. x  e.  S  ( B R x )  =  C )
17 oveq2 6081 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  ( B R x )  =  ( B R ( F `  w ) ) )
1817eqeq1d 2443 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
( B R x )  =  C  <->  ( B R ( F `  w ) )  =  C ) )
1918rspccva 3043 . . . . 5  |-  ( ( A. x  e.  S  ( B R x )  =  C  /\  ( F `  w )  e.  S )  ->  ( B R ( F `  w ) )  =  C )
2016, 19sylan 458 . . . 4  |-  ( (
ph  /\  ( F `  w )  e.  S
)  ->  ( B R ( F `  w ) )  =  C )
2114, 20syldan 457 . . 3  |-  ( (
ph  /\  w  e.  A )  ->  ( B R ( F `  w ) )  =  C )
22 fvconst2g 5937 . . . 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 2470 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( B R ( F `  w ) )  =  ( ( A  X.  { C } ) `  w ) )
251, 4, 7, 10, 12, 13, 24offveq 6317 1  |-  ( ph  ->  ( ( A  X.  { B } )  o F R F )  =  ( A  X.  { C } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   {csn 3806    X. cxp 4868    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295
This theorem is referenced by:  mbfmulc2lem  19531  i1fmulc  19587  itg1mulc  19588  itg2mulc  19631  dvcmulf  19823  coe0  20166  plymul0or  20190  expgrowth  27520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297
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