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Theorem setsabsd 12008
Description: Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)
Hypotheses
Ref Expression
setsabsd.s  |-  ( ph  ->  S  e.  V )
setsabsd.a  |-  ( ph  ->  A  e.  W )
setsabsd.b  |-  ( ph  ->  B  e.  X )
setsabsd.c  |-  ( ph  ->  C  e.  U )
Assertion
Ref Expression
setsabsd  |-  ( ph  ->  ( ( S sSet  <. A ,  B >. ) sSet  <. A ,  C >. )  =  ( S sSet  <. A ,  C >. )
)

Proof of Theorem setsabsd
StepHypRef Expression
1 setsabsd.s . . . 4  |-  ( ph  ->  S  e.  V )
2 setsabsd.a . . . 4  |-  ( ph  ->  A  e.  W )
3 setsabsd.b . . . 4  |-  ( ph  ->  B  e.  X )
4 setsresg 12007 . . . 4  |-  ( ( S  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } ) )  =  ( S  |`  ( _V  \  { A }
) ) )
51, 2, 3, 4syl3anc 1216 . . 3  |-  ( ph  ->  ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } ) )  =  ( S  |`  ( _V  \  { A }
) ) )
65uneq1d 3229 . 2  |-  ( ph  ->  ( ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } ) )  u. 
{ <. A ,  C >. } )  =  ( ( S  |`  ( _V  \  { A }
) )  u.  { <. A ,  C >. } ) )
7 setsex 12001 . . . 4  |-  ( ( S  e.  V  /\  A  e.  W  /\  B  e.  X )  ->  ( S sSet  <. A ,  B >. )  e.  _V )
81, 2, 3, 7syl3anc 1216 . . 3  |-  ( ph  ->  ( S sSet  <. A ,  B >. )  e.  _V )
9 setsabsd.c . . 3  |-  ( ph  ->  C  e.  U )
10 setsvala 12000 . . 3  |-  ( ( ( S sSet  <. A ,  B >. )  e.  _V  /\  A  e.  W  /\  C  e.  U )  ->  ( ( S sSet  <. A ,  B >. ) sSet  <. A ,  C >. )  =  ( ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } ) )  u.  { <. A ,  C >. } ) )
118, 2, 9, 10syl3anc 1216 . 2  |-  ( ph  ->  ( ( S sSet  <. A ,  B >. ) sSet  <. A ,  C >. )  =  ( ( ( S sSet  <. A ,  B >. )  |`  ( _V  \  { A } ) )  u.  { <. A ,  C >. } ) )
12 setsvala 12000 . . 3  |-  ( ( S  e.  V  /\  A  e.  W  /\  C  e.  U )  ->  ( S sSet  <. A ,  C >. )  =  ( ( S  |`  ( _V  \  { A }
) )  u.  { <. A ,  C >. } ) )
131, 2, 9, 12syl3anc 1216 . 2  |-  ( ph  ->  ( S sSet  <. A ,  C >. )  =  ( ( S  |`  ( _V  \  { A }
) )  u.  { <. A ,  C >. } ) )
146, 11, 133eqtr4d 2182 1  |-  ( ph  ->  ( ( S sSet  <. A ,  B >. ) sSet  <. A ,  C >. )  =  ( S sSet  <. A ,  C >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   _Vcvv 2686    \ cdif 3068    u. cun 3069   {csn 3527   <.cop 3530    |` cres 4541  (class class class)co 5774   sSet csts 11967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sets 11976
This theorem is referenced by: (None)
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