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Theorem 1stexg 6161
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 2748 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 fo1st 6151 . . . 4  |-  1st : _V -onto-> _V
3 fofn 5435 . . . 4  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
42, 3ax-mp 5 . . 3  |-  1st  Fn  _V
5 funfvex 5527 . . . 4  |-  ( ( Fun  1st  /\  A  e. 
dom  1st )  ->  ( 1st `  A )  e. 
_V )
65funfni 5311 . . 3  |-  ( ( 1st  Fn  _V  /\  A  e.  _V )  ->  ( 1st `  A
)  e.  _V )
74, 6mpan 424 . 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 2148   _Vcvv 2737    Fn wfn 5206   -onto->wfo 5209   ` cfv 5211   1stc1st 6132
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fo 5217  df-fv 5219  df-1st 6134
This theorem is referenced by:  elxp7  6164  xpopth  6170  eqop  6171  2nd1st  6174  2ndrn  6177  releldm2  6179  reldm  6180  dfoprab3  6185  elopabi  6189  mpofvex  6197  dfmpo  6217  cnvf1olem  6218  cnvoprab  6228  f1od2  6229  disjxp1  6230  xpmapenlem  6842  cnref1o  9626  fsumcnv  11416  fprodcnv  11604  qredeu  12067  qnumval  12155  txbas  13391  txdis  13410
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