Theorem 2ndexg 6326

Description: Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)

Assertion

Ref	Expression
2ndexg	|- ( A e. V -> ( 2nd ` A ) e. _V )

Proof of Theorem 2ndexg

Step	Hyp	Ref	Expression
1		elex 2812	. 2 |- ( A e. V -> A e. _V )
2		fo2nd 6316	. . . 4 |- 2nd : _V -onto-> _V
3		fofn 5558	. . . 4 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
4	2, 3	ax-mp 5	. . 3 |- 2nd Fn _V
5		funfvex 5652	. . . 4 |- ( ( Fun 2nd /\ A e. dom 2nd ) -> ( 2nd ` A ) e. _V )
6	5	funfni 5429	. . 3 |- ( ( 2nd Fn _V /\ A e. _V ) -> ( 2nd ` A ) e. _V )
7	4, 6	mpan 424	. 2 |- ( A e. _V -> ( 2nd ` A ) e. _V )
8	1, 7	syl 14	1 |- ( A e. V -> ( 2nd ` A ) e. _V )

Syntax hints:   -> wi 4   e. wcel 2200   _Vcvv 2800   Fn wfn 5319   -onto->wfo 5322   ` cfv 5324   2ndc2nd 6297

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fo 5330  df-fv 5332  df-2nd 6299

This theorem is referenced by:  elxp7  6328  xpopth  6334  eqop  6335  op1steq  6337  2nd1st  6338  2ndrn  6341  dfoprab3  6349  elopabi  6355  mpofvex  6365  dfmpo  6383  cnvf1olem  6384  cnvoprab  6394  f1od2  6395  elmpom  6398  xpmapenlem  7030  cc2lem  7475  cnref1o  9875  fsumcnv  11988  fprodcnv  12176  qredeu  12659  qdenval  12748  xpsff1o  13422  txbas  14972  txdis  14991  iedgvalg  15858  iedgex  15860  edgvalg  15900  wlkelvv  16146  wlk2f  16148