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Theorem 1stexg 5822
Description: Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)
Assertion
Ref Expression
1stexg  |-  ( A  e.  V  ->  ( 1st `  A )  e. 
_V )

Proof of Theorem 1stexg
StepHypRef Expression
1 elex 2583 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 fo1st 5812 . . . 4  |-  1st : _V -onto-> _V
3 fofn 5136 . . . 4  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
42, 3ax-mp 7 . . 3  |-  1st  Fn  _V
5 funfvex 5220 . . . 4  |-  ( ( Fun  1st  /\  A  e. 
dom  1st )  ->  ( 1st `  A )  e. 
_V )
65funfni 5027 . . 3  |-  ( ( 1st  Fn  _V  /\  A  e.  _V )  ->  ( 1st `  A
)  e.  _V )
74, 6mpan 408 . 2  |-  ( A  e.  _V  ->  ( 1st `  A )  e. 
_V )
81, 7syl 14 1  |-  ( A  e.  V  ->  ( 1st `  A )  e. 
_V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1409   _Vcvv 2574    Fn wfn 4925   -onto->wfo 4928   ` cfv 4930   1stc1st 5793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938  df-1st 5795
This theorem is referenced by:  elxp7  5825  xpopth  5830  eqop  5831  2nd1st  5834  2ndrn  5837  releldm2  5839  reldm  5840  dfoprab3  5845  elopabi  5849  mpt2fvex  5857  dfmpt2  5872  cnvf1olem  5873  cnvoprab  5883  f1od2  5884  cnref1o  8680
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