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Theorem 1stexg 5717
Description: Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
Assertion
Ref Expression
1stexg  V  1st ` 
_V

Proof of Theorem 1stexg
StepHypRef Expression
1 elex 2543 . 2  V  _V
2 fo1st 5707 . . . 4  1st : _V -onto-> _V
3 fofn 5033 . . . 4  1st
: _V -onto-> _V  1st 
Fn  _V
42, 3ax-mp 7 . . 3  1st  Fn  _V
5 funfvex 5117 . . . 4  Fun  1st  dom  1st  1st ` 
_V
65funfni 4925 . . 3  1st  Fn  _V  _V  1st `  _V
74, 6mpan 402 . 2  _V  1st ` 
_V
81, 7syl 14 1  V  1st ` 
_V
Colors of variables: wff set class
Syntax hints:   wi 4   wcel 1374   _Vcvv 2535    Fn wfn 4824   -onto->wfo 4827   ` cfv 4829   1stc1st 5688
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-1st 5690
This theorem is referenced by:  elxp7  5720  xpopth  5725  eqop  5726  2nd1st  5729  2ndrn  5732  releldm2  5734  reldm  5735  dfoprab3  5740  elopabi  5744  mpt2fvex  5752  dfmpt2  5767  cnvf1olem  5768
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