ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  1stvalg Unicode version

Theorem 1stvalg 5913
Description: The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
1stvalg  |-  ( A  e.  _V  ->  ( 1st `  A )  = 
U. dom  { A } )

Proof of Theorem 1stvalg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 snexg 4019 . . 3  |-  ( A  e.  _V  ->  { A }  e.  _V )
2 dmexg 4697 . . 3  |-  ( { A }  e.  _V  ->  dom  { A }  e.  _V )
3 uniexg 4265 . . 3  |-  ( dom 
{ A }  e.  _V  ->  U. dom  { A }  e.  _V )
41, 2, 33syl 17 . 2  |-  ( A  e.  _V  ->  U. dom  { A }  e.  _V )
5 sneq 3457 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
65dmeqd 4638 . . . 4  |-  ( x  =  A  ->  dom  { x }  =  dom  { A } )
76unieqd 3664 . . 3  |-  ( x  =  A  ->  U. dom  { x }  =  U. dom  { A } )
8 df-1st 5911 . . 3  |-  1st  =  ( x  e.  _V  |->  U.
dom  { x } )
97, 8fvmptg 5380 . 2  |-  ( ( A  e.  _V  /\  U.
dom  { A }  e.  _V )  ->  ( 1st `  A )  =  U. dom  { A } )
104, 9mpdan 412 1  |-  ( A  e.  _V  ->  ( 1st `  A )  = 
U. dom  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289    e. wcel 1438   _Vcvv 2619   {csn 3446   U.cuni 3653   dom cdm 4438   ` cfv 5015   1stc1st 5909
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-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fv 5023  df-1st 5911
This theorem is referenced by:  1st0  5915  op1st  5917  elxp6  5940
  Copyright terms: Public domain W3C validator