Theorem 2ndexg 5815

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 2610	. 2 |- ( A e. V -> A e. _V )
2		fo2nd 5805	. . . 4 |- 2nd : _V -onto-> _V
3		fofn 5128	. . . 4 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
4	2, 3	ax-mp 7	. . 3 |- 2nd Fn _V
5		funfvex 5212	. . . 4 |- ( ( Fun 2nd /\ A e. dom 2nd ) -> ( 2nd ` A ) e. _V )
6	5	funfni 5019	. . 3 |- ( ( 2nd Fn _V /\ A e. _V ) -> ( 2nd ` A ) e. _V )
7	4, 6	mpan 414	. 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 1433   _Vcvv 2601   Fn wfn 4917   -onto->wfo 4920   ` cfv 4922   2ndc2nd 5786

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fo 4928  df-fv 4930  df-2nd 5788

This theorem is referenced by:  elxp7  5817  xpopth  5822  eqop  5823  op1steq  5825  2nd1st  5826  2ndrn  5829  dfoprab3  5837  elopabi  5841  mpt2fvex  5849  dfmpt2  5864  cnvf1olem  5865  cnvoprab  5875  f1od2  5876  cnref1o  8733  qredeu  10479