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Theorem op1stbg 4497
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 3791 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
21inteqd 3864 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| <. A ,  B >.  =  |^| { { A } ,  { A ,  B } } )
3 snexg 4202 . . . . . 6  |-  ( A  e.  V  ->  { A }  e.  _V )
4 prexg 4229 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A ,  B }  e.  _V )
5 intprg 3892 . . . . . 6  |-  ( ( { A }  e.  _V  /\  { A ,  B }  e.  _V )  ->  |^| { { A } ,  { A ,  B } }  =  ( { A }  i^i  { A ,  B }
) )
63, 4, 5syl2an2r 595 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { { A } ,  { A ,  B } }  =  ( { A }  i^i  { A ,  B }
) )
7 snsspr1 3755 . . . . . 6  |-  { A }  C_  { A ,  B }
8 df-ss 3157 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  i^i  { A ,  B } )  =  { A } )
97, 8mpbi 145 . . . . 5  |-  ( { A }  i^i  { A ,  B }
)  =  { A }
106, 9eqtrdi 2238 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { { A } ,  { A ,  B } }  =  { A } )
112, 10eqtrd 2222 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| <. A ,  B >.  =  { A }
)
1211inteqd 3864 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| |^| <. A ,  B >.  =  |^| { A } )
13 intsng 3893 . . 3  |-  ( A  e.  V  ->  |^| { A }  =  A )
1413adantr 276 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { A }  =  A )
1512, 14eqtrd 2222 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| |^| <. A ,  B >.  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   _Vcvv 2752    i^i cin 3143    C_ wss 3144   {csn 3607   {cpr 3608   <.cop 3610   |^|cint 3859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-int 3860
This theorem is referenced by:  elxp5  5135  fundmen  6833
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