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Theorem 1stvalg 5800
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
1stvalg  |-  ( A  e.  _V  ->  ( 1st `  A )  = 
U. dom  { A } )

Proof of Theorem 1stvalg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 snexg 3964 . . 3  |-  ( A  e.  _V  ->  { A }  e.  _V )
2 dmexg 4624 . . 3  |-  ( { A }  e.  _V  ->  dom  { A }  e.  _V )
3 uniexg 4201 . . 3  |-  ( dom 
{ A }  e.  _V  ->  U. dom  { A }  e.  _V )
41, 2, 33syl 17 . 2  |-  ( A  e.  _V  ->  U. dom  { A }  e.  _V )
5 sneq 3417 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
65dmeqd 4565 . . . 4  |-  ( x  =  A  ->  dom  { x }  =  dom  { A } )
76unieqd 3620 . . 3  |-  ( x  =  A  ->  U. dom  { x }  =  U. dom  { A } )
8 df-1st 5798 . . 3  |-  1st  =  ( x  e.  _V  |->  U.
dom  { x } )
97, 8fvmptg 5280 . 2  |-  ( ( A  e.  _V  /\  U.
dom  { A }  e.  _V )  ->  ( 1st `  A )  =  U. dom  { A } )
104, 9mpdan 412 1  |-  ( A  e.  _V  ->  ( 1st `  A )  = 
U. dom  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   _Vcvv 2602   {csn 3406   U.cuni 3609   dom cdm 4371   ` cfv 4932   1stc1st 5796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-iota 4897  df-fun 4934  df-fv 4940  df-1st 5798
This theorem is referenced by:  1st0  5802  op1st  5804  elxp6  5827
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