Theorem 1stexg 6405

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 6395	. . . 4 |- 1st : _V -onto-> _V
3		fofn 5620	. . . 4 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
4	2, 3	ax-mp 5	. . 3 |- 1st Fn _V
5		funfvex 5715	. . . 4 |- ( ( Fun 1st /\ A e. dom 1st ) -> ( 1st ` A ) e. _V )
6	5	funfni 5486	. . 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 )

This proof depends on syntax axioms:   -> wi 4   e. wcel 2209   _Vcvv 2821   Fn wfn 5375   -onto->wfo 5378   ` cfv 5380   1stc1st 6376

This proof depends on 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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fo 5386  df-fv 5388  df-1st 6378

This theorem is used by:  elxp7  6408  xpopth  6414  eqop  6415  2nd1st  6418  2ndrn  6421  releldm2  6423  reldm  6424  dfoprab3  6429  elopabi  6435  mpofvex  6445  dfmpo  6463  cnvf1olem  6464  cnvoprab  6474  f1od2  6475  disjxp1  6476  elmpom  6478  xpmapenlem  7153  relresfin  7268  cnref1o  10068  fsumcnv  12229  fprodcnv  12417  qredeu  12900  qnumval  12990  xpsff1o  13729  txbas  15484  txdis  15503  vtxvalg  16462  vtxex  16464  wlkelvv  16795  wlk2f  16797