Theorem 2ndexg 6059

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

Assertion

Ref	Expression
2ndexg	⊢ (𝐴 ∈ 𝑉 → (2nd ‘𝐴) ∈ V)

Proof of Theorem 2ndexg

Step	Hyp	Ref	Expression
1		elex 2692	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		fo2nd 6049	. . . 4 ⊢ 2nd :V–onto→V
3		fofn 5342	. . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V)
4	2, 3	ax-mp 5	. . 3 ⊢ 2nd Fn V
5		funfvex 5431	. . . 4 ⊢ ((Fun 2nd ∧ 𝐴 ∈ dom 2nd ) → (2nd ‘𝐴) ∈ V)
6	5	funfni 5218	. . 3 ⊢ ((2nd Fn V ∧ 𝐴 ∈ V) → (2nd ‘𝐴) ∈ V)
7	4, 6	mpan 420	. 2 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) ∈ V)
8	1, 7	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → (2nd ‘𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1480  Vcvv 2681   Fn wfn 5113  –onto→wfo 5116  ‘cfv 5118  2^nd c2nd 6030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fo 5124  df-fv 5126  df-2nd 6032

This theorem is referenced by:  elxp7  6061  xpopth  6067  eqop  6068  op1steq  6070  2nd1st  6071  2ndrn  6074  dfoprab3  6082  elopabi  6086  mpofvex  6094  dfmpo  6113  cnvf1olem  6114  cnvoprab  6124  f1od2  6125  xpmapenlem  6736  cnref1o  9433  fsumcnv  11199  qredeu  11767  qdenval  11853  txbas  12416  txdis  12435