Theorem 1stexg 6167

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

Step	Hyp	Ref	Expression
1		elex 2748	. 2 |- ( A e. V -> A e. _V )
2		fo1st 6157	. . . 4 |- 1st : _V -onto-> _V
3		fofn 5440	. . . 4 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
4	2, 3	ax-mp 5	. . 3 |- 1st Fn _V
5		funfvex 5532	. . . 4 |- ( ( Fun 1st /\ A e. dom 1st ) -> ( 1st ` A ) e. _V )
6	5	funfni 5316	. . 3 |- ( ( 1st Fn _V /\ A e. _V ) -> ( 1st ` A ) e. _V )
7	4, 6	mpan 424	. 2 |- ( A e. _V -> ( 1st ` A ) e. _V )
8	1, 7	syl 14	1 |- ( A e. V -> ( 1st ` A ) e. _V )

Syntax hints:   -> wi 4   e. wcel 2148   _Vcvv 2737   Fn wfn 5211   -onto->wfo 5214   ` cfv 5216   1stc1st 6138

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 4121  ax-pow 4174  ax-pr 4209  ax-un 4433

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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fo 5222  df-fv 5224  df-1st 6140

This theorem is referenced by:  elxp7  6170  xpopth  6176  eqop  6177  2nd1st  6180  2ndrn  6183  releldm2  6185  reldm  6186  dfoprab3  6191  elopabi  6195  mpofvex  6203  dfmpo  6223  cnvf1olem  6224  cnvoprab  6234  f1od2  6235  disjxp1  6236  xpmapenlem  6848  cnref1o  9649  fsumcnv  11444  fprodcnv  11632  qredeu  12096  qnumval  12184  xpsff1o  12767  txbas  13728  txdis  13747