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Theorem setsabs 14320
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 14319 . . . 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 3616 . 2  |-  ( ( S  e.  V  /\  C  e.  W )  ->  ( ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } ) )  u. 
{ <. A ,  C >. } )  =  ( ( S  |`  ( _V  \  { A }
) )  u.  { <. A ,  C >. } ) )
4 ovex 6224 . . . 4  |-  ( S sSet  <. A ,  B >. )  e.  _V
54a1i 11 . . 3  |-  ( S  e.  V  ->  ( S sSet  <. A ,  B >. )  e.  _V )
6 setsval 14315 . . 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 14315 . 2  |-  ( ( S  e.  V  /\  C  e.  W )  ->  ( S sSet  <. A ,  C >. )  =  ( ( S  |`  ( _V  \  { A }
) )  u.  { <. A ,  C >. } ) )
93, 7, 83eqtr4d 2505 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 1370    e. wcel 1758   _Vcvv 3076    \ cdif 3432    u. cun 3433   {csn 3984   <.cop 3990    |` cres 4949  (class class class)co 6199   sSet csts 14289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638  ax-un 6481
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-res 4959  df-iota 5488  df-fun 5527  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-sets 14297
This theorem is referenced by:  ressress  14353  rescabs  14864
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