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Theorem caofid2 6267
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 5564 . . 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 5524 . . 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 5564 . . 3  |-  ( C  e.  X  ->  ( A  X.  { C }
)  Fn  A )
108, 9syl 16 . 2  |-  ( ph  ->  ( A  X.  { C } )  Fn  A
)
11 fvconst2g 5877 . . 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 2381 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
145ffvelrnda 5802 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
15 caofid2.5 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  ( B R x )  =  C )
1615ralrimiva 2725 . . . . 5  |-  ( ph  ->  A. x  e.  S  ( B R x )  =  C )
17 oveq2 6021 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  ( B R x )  =  ( B R ( F `  w ) ) )
1817eqeq1d 2388 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
( B R x )  =  C  <->  ( B R ( F `  w ) )  =  C ) )
1918rspccva 2987 . . . . 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 5877 . . . 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 2415 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( B R ( F `  w ) )  =  ( ( A  X.  { C } ) `  w ) )
251, 4, 7, 10, 12, 13, 24offveq 6257 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 1649    e. wcel 1717   A.wral 2642   {csn 3750    X. cxp 4809    Fn wfn 5382   -->wf 5383   ` cfv 5387  (class class class)co 6013    o Fcof 6235
This theorem is referenced by:  mbfmulc2lem  19399  i1fmulc  19455  itg1mulc  19456  itg2mulc  19499  dvcmulf  19691  coe0  20034  plymul0or  20058  expgrowth  27214
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237
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