Theorem 2ndexg 5977

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 2644	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		fo2nd 5967	. . . 4 ⊢ 2nd :V–onto→V
3		fofn 5270	. . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V)
4	2, 3	ax-mp 7	. . 3 ⊢ 2nd Fn V
5		funfvex 5357	. . . 4 ⊢ ((Fun 2nd ∧ 𝐴 ∈ dom 2nd ) → (2nd ‘𝐴) ∈ V)
6	5	funfni 5148	. . 3 ⊢ ((2nd Fn V ∧ 𝐴 ∈ V) → (2nd ‘𝐴) ∈ V)
7	4, 6	mpan 416	. 2 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) ∈ V)
8	1, 7	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → (2nd ‘𝐴) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1445  Vcvv 2633   Fn wfn 5044  –onto→wfo 5047  ‘cfv 5049  2^nd c2nd 5948

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fo 5055  df-fv 5057  df-2nd 5950

This theorem is referenced by:  elxp7  5979  xpopth  5984  eqop  5985  op1steq  5987  2nd1st  5988  2ndrn  5991  dfoprab3  5999  elopabi  6003  mpt2fvex  6011  dfmpt2  6026  cnvf1olem  6027  cnvoprab  6037  f1od2  6038  xpmapenlem  6645  djur  6837  cnref1o  9232  fsumcnv  10996  qredeu  11522  qdenval  11607