MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1stval Unicode version

Theorem 1stval 6211
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
1stval  |-  ( 1st `  A )  =  U. dom  { A }

Proof of Theorem 1stval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sneq 3727 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
21dmeqd 4963 . . . 4  |-  ( x  =  A  ->  dom  { x }  =  dom  { A } )
32unieqd 3919 . . 3  |-  ( x  =  A  ->  U. dom  { x }  =  U. dom  { A } )
4 df-1st 6209 . . 3  |-  1st  =  ( x  e.  _V  |->  U.
dom  { x } )
5 snex 4297 . . . . 5  |-  { A }  e.  _V
65dmex 5023 . . . 4  |-  dom  { A }  e.  _V
76uniex 4598 . . 3  |-  U. dom  { A }  e.  _V
83, 4, 7fvmpt 5685 . 2  |-  ( A  e.  _V  ->  ( 1st `  A )  = 
U. dom  { A } )
9 fvprc 5602 . . 3  |-  ( -.  A  e.  _V  ->  ( 1st `  A )  =  (/) )
10 snprc 3771 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1110biimpi 186 . . . . . . 7  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1211dmeqd 4963 . . . . . 6  |-  ( -.  A  e.  _V  ->  dom 
{ A }  =  dom  (/) )
13 dm0 4974 . . . . . 6  |-  dom  (/)  =  (/)
1412, 13syl6eq 2406 . . . . 5  |-  ( -.  A  e.  _V  ->  dom 
{ A }  =  (/) )
1514unieqd 3919 . . . 4  |-  ( -.  A  e.  _V  ->  U.
dom  { A }  =  U. (/) )
16 uni0 3935 . . . 4  |-  U. (/)  =  (/)
1715, 16syl6eq 2406 . . 3  |-  ( -.  A  e.  _V  ->  U.
dom  { A }  =  (/) )
189, 17eqtr4d 2393 . 2  |-  ( -.  A  e.  _V  ->  ( 1st `  A )  =  U. dom  { A } )
198, 18pm2.61i 156 1  |-  ( 1st `  A )  =  U. dom  { A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1642    e. wcel 1710   _Vcvv 2864   (/)c0 3531   {csn 3716   U.cuni 3908   dom cdm 4771   ` cfv 5337   1stc1st 6207
This theorem is referenced by:  1st0  6213  op1st  6215  1st2val  6232  elxp6  6238  1stnpr  23297  mpt2xopxnop0  27438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-iota 5301  df-fun 5339  df-fv 5345  df-1st 6209
  Copyright terms: Public domain W3C validator