Theorem 1stexg 6391

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 2833	. 2 |- ( A e. V -> A e. _V )
2		fo1st 6381	. . . 4 |- 1st : _V -onto-> _V
3		fofn 5612	. . . 4 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
4	2, 3	ax-mp 5	. . 3 |- 1st Fn _V
5		funfvex 5707	. . . 4 |- ( ( Fun 1st /\ A e. dom 1st ) -> ( 1st ` A ) e. _V )
6	5	funfni 5478	. . 3 |- ( ( 1st Fn _V /\ A e. _V ) -> ( 1st ` A ) e. _V )
7	4, 6	mpan 428	. 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 2209   _Vcvv 2821   Fn wfn 5367   -onto->wfo 5370   ` cfv 5372   1stc1st 6362

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-1st 6364

This theorem is referenced by:  elxp7  6394  xpopth  6400  eqop  6401  2nd1st  6404  2ndrn  6407  releldm2  6409  reldm  6410  dfoprab3  6415  elopabi  6421  mpofvex  6431  dfmpo  6449  cnvf1olem  6450  cnvoprab  6460  f1od2  6461  disjxp1  6462  elmpom  6464  xpmapenlem  7139  cnref1o  10030  fsumcnv  12182  fprodcnv  12370  qredeu  12853  qnumval  12941  xpsff1o  13647  txbas  15282  txdis  15301  vtxvalg  16171  vtxex  16173  wlkelvv  16504  wlk2f  16506