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Theorem setsabs 14195
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 14194 . . . 4  |-  ( S  e.  V  ->  (
( S sSet  <. A ,  B >. )  |`  ( _V  \  { A }
) )  =  ( S  |`  ( _V  \  { A } ) ) )
21adantr 465 . . 3  |-  ( ( S  e.  V  /\  C  e.  W )  ->  ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } ) )  =  ( S  |`  ( _V  \  { A }
) ) )
32uneq1d 3504 . 2  |-  ( ( S  e.  V  /\  C  e.  W )  ->  ( ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } ) )  u. 
{ <. A ,  C >. } )  =  ( ( S  |`  ( _V  \  { A }
) )  u.  { <. A ,  C >. } ) )
4 ovex 6111 . . . 4  |-  ( S sSet  <. A ,  B >. )  e.  _V
54a1i 11 . . 3  |-  ( S  e.  V  ->  ( S sSet  <. A ,  B >. )  e.  _V )
6 setsval 14190 . . 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 471 . 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 14190 . 2  |-  ( ( S  e.  V  /\  C  e.  W )  ->  ( S sSet  <. A ,  C >. )  =  ( ( S  |`  ( _V  \  { A }
) )  u.  { <. A ,  C >. } ) )
93, 7, 83eqtr4d 2480 1  |-  ( ( S  e.  V  /\  C  e.  W )  ->  ( ( S sSet  <. A ,  B >. ) sSet  <. A ,  C >. )  =  ( S sSet  <. A ,  C >. )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2967    \ cdif 3320    u. cun 3321   {csn 3872   <.cop 3878    |` cres 4837  (class class class)co 6086   sSet csts 14164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-res 4847  df-iota 5376  df-fun 5415  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-sets 14172
This theorem is referenced by:  ressress  14227  rescabs  14738
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