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Theorem 2ndexg 6069
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 2697 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 fo2nd 6059 . . . 4  |-  2nd : _V -onto-> _V
3 fofn 5350 . . . 4  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
42, 3ax-mp 5 . . 3  |-  2nd  Fn  _V
5 funfvex 5441 . . . 4  |-  ( ( Fun  2nd  /\  A  e. 
dom  2nd )  ->  ( 2nd `  A )  e. 
_V )
65funfni 5226 . . 3  |-  ( ( 2nd  Fn  _V  /\  A  e.  _V )  ->  ( 2nd `  A
)  e.  _V )
74, 6mpan 420 . 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 1480   _Vcvv 2686    Fn wfn 5121   -onto->wfo 5124   ` cfv 5126   2ndc2nd 6040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134  ax-un 4358
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-br 3933  df-opab 3993  df-mpt 3994  df-id 4218  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-fo 5132  df-fv 5134  df-2nd 6042
This theorem is referenced by:  elxp7  6071  xpopth  6077  eqop  6078  op1steq  6080  2nd1st  6081  2ndrn  6084  dfoprab3  6092  elopabi  6096  mpofvex  6104  dfmpo  6123  cnvf1olem  6124  cnvoprab  6134  f1od2  6135  xpmapenlem  6746  cc2lem  7093  cnref1o  9462  fsumcnv  11230  qredeu  11801  qdenval  11887  txbas  12453  txdis  12472
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