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Theorem op1stbg 4476
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 3774 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  W )  -> 
<. A ,  B >.  =  { { A } ,  { A ,  B } } )
21inteqd 3847 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| <. A ,  B >.  =  |^| { { A } ,  { A ,  B } } )
3 snexg 4181 . . . . . 6  |-  ( A  e.  V  ->  { A }  e.  _V )
4 prexg 4208 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  { A ,  B }  e.  _V )
5 intprg 3875 . . . . . 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 3739 . . . . . 6  |-  { A }  C_  { A ,  B }
8 df-ss 3142 . . . . . 6  |-  ( { A }  C_  { A ,  B }  <->  ( { A }  i^i  { A ,  B } )  =  { A } )
97, 8mpbi 145 . . . . 5  |-  ( { A }  i^i  { A ,  B }
)  =  { A }
106, 9eqtrdi 2226 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { { A } ,  { A ,  B } }  =  { A } )
112, 10eqtrd 2210 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| <. A ,  B >.  =  { A }
)
1211inteqd 3847 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| |^| <. A ,  B >.  =  |^| { A } )
13 intsng 3876 . . 3  |-  ( A  e.  V  ->  |^| { A }  =  A )
1413adantr 276 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| { A }  =  A )
1512, 14eqtrd 2210 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  |^| |^| <. A ,  B >.  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   _Vcvv 2737    i^i cin 3128    C_ wss 3129   {csn 3591   {cpr 3592   <.cop 3594   |^|cint 3842
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-int 3843
This theorem is referenced by:  elxp5  5113  fundmen  6800
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