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Theorem 1stval 6343
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 3817 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
21dmeqd 5064 . . . 4  |-  ( x  =  A  ->  dom  { x }  =  dom  { A } )
32unieqd 4018 . . 3  |-  ( x  =  A  ->  U. dom  { x }  =  U. dom  { A } )
4 df-1st 6341 . . 3  |-  1st  =  ( x  e.  _V  |->  U.
dom  { x } )
5 snex 4397 . . . . 5  |-  { A }  e.  _V
65dmex 5124 . . . 4  |-  dom  { A }  e.  _V
76uniex 4697 . . 3  |-  U. dom  { A }  e.  _V
83, 4, 7fvmpt 5798 . 2  |-  ( A  e.  _V  ->  ( 1st `  A )  = 
U. dom  { A } )
9 fvprc 5714 . . 3  |-  ( -.  A  e.  _V  ->  ( 1st `  A )  =  (/) )
10 snprc 3863 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1110biimpi 187 . . . . . . 7  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1211dmeqd 5064 . . . . . 6  |-  ( -.  A  e.  _V  ->  dom 
{ A }  =  dom  (/) )
13 dm0 5075 . . . . . 6  |-  dom  (/)  =  (/)
1412, 13syl6eq 2483 . . . . 5  |-  ( -.  A  e.  _V  ->  dom 
{ A }  =  (/) )
1514unieqd 4018 . . . 4  |-  ( -.  A  e.  _V  ->  U.
dom  { A }  =  U. (/) )
16 uni0 4034 . . . 4  |-  U. (/)  =  (/)
1715, 16syl6eq 2483 . . 3  |-  ( -.  A  e.  _V  ->  U.
dom  { A }  =  (/) )
189, 17eqtr4d 2470 . 2  |-  ( -.  A  e.  _V  ->  ( 1st `  A )  =  U. dom  { A } )
198, 18pm2.61i 158 1  |-  ( 1st `  A )  =  U. dom  { A }
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620   {csn 3806   U.cuni 4007   dom cdm 4870   ` cfv 5446   1stc1st 6339
This theorem is referenced by:  1st0  6345  op1st  6347  1st2val  6364  elxp6  6370  mpt2xopxnop0  6458  1stnpr  24085
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-1st 6341
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