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Theorem 1stval 6795
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 3978 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
21dmeqd 5037 . . . 4  |-  ( x  =  A  ->  dom  { x }  =  dom  { A } )
32unieqd 4208 . . 3  |-  ( x  =  A  ->  U. dom  { x }  =  U. dom  { A } )
4 df-1st 6793 . . 3  |-  1st  =  ( x  e.  _V  |->  U.
dom  { x } )
5 snex 4641 . . . . 5  |-  { A }  e.  _V
65dmex 6726 . . . 4  |-  dom  { A }  e.  _V
76uniex 6587 . . 3  |-  U. dom  { A }  e.  _V
83, 4, 7fvmpt 5948 . 2  |-  ( A  e.  _V  ->  ( 1st `  A )  = 
U. dom  { A } )
9 fvprc 5859 . . 3  |-  ( -.  A  e.  _V  ->  ( 1st `  A )  =  (/) )
10 snprc 4035 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1110biimpi 198 . . . . . . 7  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1211dmeqd 5037 . . . . . 6  |-  ( -.  A  e.  _V  ->  dom 
{ A }  =  dom  (/) )
13 dm0 5048 . . . . . 6  |-  dom  (/)  =  (/)
1412, 13syl6eq 2501 . . . . 5  |-  ( -.  A  e.  _V  ->  dom 
{ A }  =  (/) )
1514unieqd 4208 . . . 4  |-  ( -.  A  e.  _V  ->  U.
dom  { A }  =  U. (/) )
16 uni0 4225 . . . 4  |-  U. (/)  =  (/)
1715, 16syl6eq 2501 . . 3  |-  ( -.  A  e.  _V  ->  U.
dom  { A }  =  (/) )
189, 17eqtr4d 2488 . 2  |-  ( -.  A  e.  _V  ->  ( 1st `  A )  =  U. dom  { A } )
198, 18pm2.61i 168 1  |-  ( 1st `  A )  =  U. dom  { A }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1444    e. wcel 1887   _Vcvv 3045   (/)c0 3731   {csn 3968   U.cuni 4198   dom cdm 4834   ` cfv 5582   1stc1st 6791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fv 5590  df-1st 6793
This theorem is referenced by:  1stnpr  6797  1st0  6799  op1st  6801  1st2val  6819  elxp6  6825  mpt2xopxnop0  6961
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