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Theorem 2ndexg 6192
Description: Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
Assertion
Ref Expression
2ndexg  |-  ( A  e.  V  ->  ( 2nd `  A )  e. 
_V )

Proof of Theorem 2ndexg
StepHypRef Expression
1 elex 2763 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 fo2nd 6182 . . . 4  |-  2nd : _V -onto-> _V
3 fofn 5459 . . . 4  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
42, 3ax-mp 5 . . 3  |-  2nd  Fn  _V
5 funfvex 5551 . . . 4  |-  ( ( Fun  2nd  /\  A  e. 
dom  2nd )  ->  ( 2nd `  A )  e. 
_V )
65funfni 5335 . . 3  |-  ( ( 2nd  Fn  _V  /\  A  e.  _V )  ->  ( 2nd `  A
)  e.  _V )
74, 6mpan 424 . 2  |-  ( A  e.  _V  ->  ( 2nd `  A )  e. 
_V )
81, 7syl 14 1  |-  ( A  e.  V  ->  ( 2nd `  A )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2160   _Vcvv 2752    Fn wfn 5230   -onto->wfo 5233   ` cfv 5235   2ndc2nd 6163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fo 5241  df-fv 5243  df-2nd 6165
This theorem is referenced by:  elxp7  6194  xpopth  6200  eqop  6201  op1steq  6203  2nd1st  6204  2ndrn  6207  dfoprab3  6215  elopabi  6219  mpofvex  6227  dfmpo  6247  cnvf1olem  6248  cnvoprab  6258  f1od2  6259  xpmapenlem  6876  cc2lem  7294  cnref1o  9679  fsumcnv  11476  fprodcnv  11664  qredeu  12128  qdenval  12217  xpsff1o  12822  txbas  14210  txdis  14229
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