Theorem 2ndexg 6331

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 2814	. 2 |- ( A e. V -> A e. _V )
2		fo2nd 6321	. . . 4 |- 2nd : _V -onto-> _V
3		fofn 5561	. . . 4 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
4	2, 3	ax-mp 5	. . 3 |- 2nd Fn _V
5		funfvex 5656	. . . 4 |- ( ( Fun 2nd /\ A e. dom 2nd ) -> ( 2nd ` A ) e. _V )
6	5	funfni 5432	. . 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 2202   _Vcvv 2802   Fn wfn 5321   -onto->wfo 5324   ` cfv 5326   2ndc2nd 6302

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-2nd 6304

This theorem is referenced by:  elxp7  6333  xpopth  6339  eqop  6340  op1steq  6342  2nd1st  6343  2ndrn  6346  dfoprab3  6354  elopabi  6360  mpofvex  6370  dfmpo  6388  cnvf1olem  6389  cnvoprab  6399  f1od2  6400  elmpom  6403  xpmapenlem  7035  cc2lem  7485  cnref1o  9885  fsumcnv  12016  fprodcnv  12204  qredeu  12687  qdenval  12776  xpsff1o  13450  txbas  15001  txdis  15020  iedgvalg  15887  iedgex  15889  edgvalg  15929  wlkelvv  16219  wlk2f  16221