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Theorem caofid2 6548
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 5766 . . 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 5724 . . 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 5766 . . 3  |-  ( C  e.  X  ->  ( A  X.  { C }
)  Fn  A )
108, 9syl 16 . 2  |-  ( ph  ->  ( A  X.  { C } )  Fn  A
)
11 fvconst2g 6107 . . 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 2463 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  =  ( F `  w ) )
145ffvelrnda 6014 . . . 4  |-  ( (
ph  /\  w  e.  A )  ->  ( F `  w )  e.  S )
15 caofid2.5 . . . . . 6  |-  ( (
ph  /\  x  e.  S )  ->  ( B R x )  =  C )
1615ralrimiva 2873 . . . . 5  |-  ( ph  ->  A. x  e.  S  ( B R x )  =  C )
17 oveq2 6285 . . . . . . 7  |-  ( x  =  ( F `  w )  ->  ( B R x )  =  ( B R ( F `  w ) ) )
1817eqeq1d 2464 . . . . . 6  |-  ( x  =  ( F `  w )  ->  (
( B R x )  =  C  <->  ( B R ( F `  w ) )  =  C ) )
1918rspccva 3208 . . . . 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 6107 . . . 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 2506 . 2  |-  ( (
ph  /\  w  e.  A )  ->  ( B R ( F `  w ) )  =  ( ( A  X.  { C } ) `  w ) )
251, 4, 7, 10, 12, 13, 24offveq 6538 1  |-  ( ph  ->  ( ( A  X.  { B } )  oF R F )  =  ( A  X.  { C } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809   {csn 4022    X. cxp 4992    Fn wfn 5576   -->wf 5577   ` cfv 5581  (class class class)co 6277    oFcof 6515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517
This theorem is referenced by:  mbfmulc2lem  21784  i1fmulc  21840  itg1mulc  21841  itg2mulc  21884  dvcmulf  22078  coe0  22382  plymul0or  22406  expgrowth  30797
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