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Theorem op1stbg 4291
Description: Extract the first member of an ordered pair. Theorem 73 of [Suppes] p. 42. (Contributed by Jim Kingdon, 17-Dec-2018.)
Assertion
Ref Expression
op1stbg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| |^| <. A ,  B >.  =  A )

Proof of Theorem op1stbg
StepHypRef Expression
1 dfopg 3615 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
21inteqd 3688 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| <. A ,  B >.  =  |^| { { A } ,  { A ,  B } } )
3 snexg 4010 . . . . . 6  |-  ( A  e.  V  ->  { A }  e.  _V )
4 prexg 4029 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A ,  B }  e.  _V )
5 intprg 3716 . . . . . 6  |-  ( ( { A }  e.  _V  /\  { A ,  B }  e.  _V )  ->  |^| { { A } ,  { A ,  B } }  =  ( { A }  i^i  { A ,  B }
) )
63, 4, 5syl2an2r 562 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { { A } ,  { A ,  B } }  =  ( { A }  i^i  { A ,  B }
) )
7 snsspr1 3580 . . . . . 6  |-  { A }  C_  { A ,  B }
8 df-ss 3010 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  i^i  { A ,  B } )  =  { A } )
97, 8mpbi 143 . . . . 5  |-  ( { A }  i^i  { A ,  B }
)  =  { A }
106, 9syl6eq 2136 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { { A } ,  { A ,  B } }  =  { A } )
112, 10eqtrd 2120 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| <. A ,  B >.  =  { A }
)
1211inteqd 3688 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| |^| <. A ,  B >.  =  |^| { A } )
13 intsng 3717 . . 3  |-  ( A  e.  V  ->  |^| { A }  =  A )
1413adantr 270 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { A }  =  A )
1512, 14eqtrd 2120 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| |^| <. A ,  B >.  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   _Vcvv 2619    i^i cin 2996    C_ wss 2997   {csn 3441   {cpr 3442   <.cop 3444   |^|cint 3683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-int 3684
This theorem is referenced by:  elxp5  4906  fundmen  6503
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