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Theorem setsabs 13488
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 13487 . . . 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 3492 . 2  |-  ( ( S  e.  V  /\  C  e.  W )  ->  ( ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } ) )  u. 
{ <. A ,  C >. } )  =  ( ( S  |`  ( _V  \  { A }
) )  u.  { <. A ,  C >. } ) )
4 ovex 6098 . . . 4  |-  ( S sSet  <. A ,  B >. )  e.  _V
54a1i 11 . . 3  |-  ( S  e.  V  ->  ( S sSet  <. A ,  B >. )  e.  _V )
6 setsval 13485 . . 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 13485 . 2  |-  ( ( S  e.  V  /\  C  e.  W )  ->  ( S sSet  <. A ,  C >. )  =  ( ( S  |`  ( _V  \  { A }
) )  u.  { <. A ,  C >. } ) )
93, 7, 83eqtr4d 2477 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 1652    e. wcel 1725   _Vcvv 2948    \ cdif 3309    u. cun 3310   {csn 3806   <.cop 3809    |` cres 4872  (class class class)co 6073   sSet csts 13459
This theorem is referenced by:  ressress  13518  rescabs  14025
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-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
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-rab 2706  df-v 2950  df-sbc 3154  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-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-sets 13467
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