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Theorem 1stval 6310
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 3785 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
21dmeqd 5031 . . . 4  |-  ( x  =  A  ->  dom  { x }  =  dom  { A } )
32unieqd 3986 . . 3  |-  ( x  =  A  ->  U. dom  { x }  =  U. dom  { A } )
4 df-1st 6308 . . 3  |-  1st  =  ( x  e.  _V  |->  U.
dom  { x } )
5 snex 4365 . . . . 5  |-  { A }  e.  _V
65dmex 5091 . . . 4  |-  dom  { A }  e.  _V
76uniex 4664 . . 3  |-  U. dom  { A }  e.  _V
83, 4, 7fvmpt 5765 . 2  |-  ( A  e.  _V  ->  ( 1st `  A )  = 
U. dom  { A } )
9 fvprc 5681 . . 3  |-  ( -.  A  e.  _V  ->  ( 1st `  A )  =  (/) )
10 snprc 3831 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
1110biimpi 187 . . . . . . 7  |-  ( -.  A  e.  _V  ->  { A }  =  (/) )
1211dmeqd 5031 . . . . . 6  |-  ( -.  A  e.  _V  ->  dom 
{ A }  =  dom  (/) )
13 dm0 5042 . . . . . 6  |-  dom  (/)  =  (/)
1412, 13syl6eq 2452 . . . . 5  |-  ( -.  A  e.  _V  ->  dom 
{ A }  =  (/) )
1514unieqd 3986 . . . 4  |-  ( -.  A  e.  _V  ->  U.
dom  { A }  =  U. (/) )
16 uni0 4002 . . . 4  |-  U. (/)  =  (/)
1715, 16syl6eq 2452 . . 3  |-  ( -.  A  e.  _V  ->  U.
dom  { A }  =  (/) )
189, 17eqtr4d 2439 . 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 1649    e. wcel 1721   _Vcvv 2916   (/)c0 3588   {csn 3774   U.cuni 3975   dom cdm 4837   ` cfv 5413   1stc1st 6306
This theorem is referenced by:  1st0  6312  op1st  6314  1st2val  6331  elxp6  6337  mpt2xopxnop0  6425  1stnpr  24046
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5377  df-fun 5415  df-fv 5421  df-1st 6308
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