MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  setsabs Unicode version

Theorem setsabs 13451
Description: Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.)
Assertion
Ref Expression
setsabs  |-  ( ( S  e.  V  /\  C  e.  W )  ->  ( ( S sSet  <. A ,  B >. ) sSet  <. A ,  C >. )  =  ( S sSet  <. A ,  C >. )
)

Proof of Theorem setsabs
StepHypRef Expression
1 setsres 13450 . . . 4  |-  ( S  e.  V  ->  (
( S sSet  <. A ,  B >. )  |`  ( _V  \  { A }
) )  =  ( S  |`  ( _V  \  { A } ) ) )
21adantr 452 . . 3  |-  ( ( S  e.  V  /\  C  e.  W )  ->  ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } ) )  =  ( S  |`  ( _V  \  { A }
) ) )
32uneq1d 3460 . 2  |-  ( ( S  e.  V  /\  C  e.  W )  ->  ( ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } ) )  u. 
{ <. A ,  C >. } )  =  ( ( S  |`  ( _V  \  { A }
) )  u.  { <. A ,  C >. } ) )
4 ovex 6065 . . . 4  |-  ( S sSet  <. A ,  B >. )  e.  _V
54a1i 11 . . 3  |-  ( S  e.  V  ->  ( S sSet  <. A ,  B >. )  e.  _V )
6 setsval 13448 . . 3  |-  ( ( ( S sSet  <. A ,  B >. )  e.  _V  /\  C  e.  W )  ->  ( ( S sSet  <. A ,  B >. ) sSet  <. A ,  C >. )  =  ( ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } ) )  u.  { <. A ,  C >. } ) )
75, 6sylan 458 . 2  |-  ( ( S  e.  V  /\  C  e.  W )  ->  ( ( S sSet  <. A ,  B >. ) sSet  <. A ,  C >. )  =  ( ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } ) )  u.  { <. A ,  C >. } ) )
8 setsval 13448 . 2  |-  ( ( S  e.  V  /\  C  e.  W )  ->  ( S sSet  <. A ,  C >. )  =  ( ( S  |`  ( _V  \  { A }
) )  u.  { <. A ,  C >. } ) )
93, 7, 83eqtr4d 2446 1  |-  ( ( S  e.  V  /\  C  e.  W )  ->  ( ( S sSet  <. A ,  B >. ) sSet  <. A ,  C >. )  =  ( S sSet  <. A ,  C >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    \ cdif 3277    u. cun 3278   {csn 3774   <.cop 3777    |` cres 4839  (class class class)co 6040   sSet csts 13422
This theorem is referenced by:  ressress  13481  rescabs  13988
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-res 4849  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-sets 13430
  Copyright terms: Public domain W3C validator