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Theorem 1stval 6354
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 3827 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
21dmeqd 5075 . . . 4  |-  ( x  =  A  ->  dom  { x }  =  dom  { A } )
32unieqd 4028 . . 3  |-  ( x  =  A  ->  U. dom  { x }  =  U. dom  { A } )
4 df-1st 6352 . . 3  |-  1st  =  ( x  e.  _V  |->  U.
dom  { x } )
5 snex 4408 . . . . 5  |-  { A }  e.  _V
65dmex 5135 . . . 4  |-  dom  { A }  e.  _V
76uniex 4708 . . 3  |-  U. dom  { A }  e.  _V
83, 4, 7fvmpt 5809 . 2  |-  ( A  e.  _V  ->  ( 1st `  A )  = 
U. dom  { A } )
9 fvprc 5725 . . 3  |-  ( -.  A  e.  _V  ->  ( 1st `  A )  =  (/) )
10 snprc 3873 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1110biimpi 188 . . . . . . 7  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1211dmeqd 5075 . . . . . 6  |-  ( -.  A  e.  _V  ->  dom 
{ A }  =  dom  (/) )
13 dm0 5086 . . . . . 6  |-  dom  (/)  =  (/)
1412, 13syl6eq 2486 . . . . 5  |-  ( -.  A  e.  _V  ->  dom 
{ A }  =  (/) )
1514unieqd 4028 . . . 4  |-  ( -.  A  e.  _V  ->  U.
dom  { A }  =  U. (/) )
16 uni0 4044 . . . 4  |-  U. (/)  =  (/)
1715, 16syl6eq 2486 . . 3  |-  ( -.  A  e.  _V  ->  U.
dom  { A }  =  (/) )
189, 17eqtr4d 2473 . 2  |-  ( -.  A  e.  _V  ->  ( 1st `  A )  =  U. dom  { A } )
198, 18pm2.61i 159 1  |-  ( 1st `  A )  =  U. dom  { A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1653    e. wcel 1726   _Vcvv 2958   (/)c0 3630   {csn 3816   U.cuni 4017   dom cdm 4881   ` cfv 5457   1stc1st 6350
This theorem is referenced by:  1st0  6356  op1st  6358  1st2val  6375  elxp6  6381  mpt2xopxnop0  6469  1stnpr  24098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fv 5465  df-1st 6352
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