Theorem 2ndexg 5847

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 2619	. 2 |- ( A e. V -> A e. _V )
2		fo2nd 5837	. . . 4 |- 2nd : _V -onto-> _V
3		fofn 5160	. . . 4 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
4	2, 3	ax-mp 7	. . 3 |- 2nd Fn _V
5		funfvex 5244	. . . 4 |- ( ( Fun 2nd /\ A e. dom 2nd ) -> ( 2nd ` A ) e. _V )
6	5	funfni 5051	. . 3 |- ( ( 2nd Fn _V /\ A e. _V ) -> ( 2nd ` A ) e. _V )
7	4, 6	mpan 415	. 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 1434   _Vcvv 2610   Fn wfn 4948   -onto->wfo 4951   ` cfv 4953   2ndc2nd 5818

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2826  df-un 2987  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-br 3807  df-opab 3861  df-mpt 3862  df-id 4077  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-fo 4959  df-fv 4961  df-2nd 5820

This theorem is referenced by:  elxp7  5849  xpopth  5854  eqop  5855  op1steq  5857  2nd1st  5858  2ndrn  5861  dfoprab3  5869  elopabi  5873  mpt2fvex  5881  dfmpt2  5896  cnvf1olem  5897  cnvoprab  5907  f1od2  5908  cnref1o  8850  qredeu  10670  qdenval  10755